Elements Of Applied Mathematics 4j1f2x

-3

-2

p

-!

-frl

is

called its absolute value.

**Z

+4

*?

-*oo

Fig. 51

Beginning at the extreme the right, numbers

— oo

may

left

and ing constantly

to

be said to increase continuously from

+00. Beginning at the extreme right and left, numbers may be said to decrease continuously from + 0° through Beginning at any to — 00 point and ing to the right gives increasing numbers, while through

to

ing constantly to the

.

ing to the left gives decreasing numbers. 62. Variables.

A

variable

is

a number which changes and

es through a series of successive

values.

through the whole scale of values from

may

— 00

may

It

to

+

through a certain portion of the scale only.

variable

is

confined to a portion of the

A

number b, if it

is

+

it

If the

number system,

from the position — 15 in the scale to the position said to have the interval — 15 to + 6. a to

or

00,

6,

it'

as is

said to vary continuously in a given interval,

starts

with the value a and increases (or decreases) 91

:

APPLIED MATHEMATICS

92

way

to the value b in such a

a and

b (integral, fractional,

assume

as to

and

all

values between

irrational) in the order of

their magnitude.

numbers. One number is greater than a a positive number must be added to the second to produce the first. Thus 3 is greater than 8, since + 5 must 63. Inequality of

second

if

—

be added to

—

—

— 3.

8 to obtain

One number

than a second

is less

if

a positive number must

first. Thus — 17 must be subtracted from — 12 to

be subtracted from the second to obtain the is less

obtain

than

—

—

+5

12, since

17.

The relation Thus - 3 > -

of inequality

10

>

-

is

usually expressed by a symbol.

< - 12, 2 < 7. a variable. The value of an expression involving a variable depends upon the value of the variable. 8,

4,

17

64. Function of

The expression

x

2

—1

is called a function of the variable. Thus a function of x (written f(x) = x 2 — 1, and read

is

" function of

—

2,-1,

0,

-

+1,

0,

+2

—

2

x equals

a;

1 "), for

respectively,

x2

when x

—

has the values

1 has the values 3,

1, 0, 3.

The

variable to which

the independent variable

depends upon

it

;

we may

give values at will

is

called

but the expression or variable which

for its value is called the dependent variable,

The volume of a cube is a function of the edge, 3 The area of a circle is a function of the radius, e 2 The distance through which a body falls is a 7rr

or function.

v

a

= f{e) = = f(r) =

.

.

function of the time,

s

= f(i) =

and dependent variables Exercise. Plot the

£ gf.

graph of the function 2 x 3

Give x integral values from

—

3 to 4

-

2z 3

-3x2 -12x +

4

Name

-41

the independent

in the preceding illustrations.

— 3 x — 12 x + 4. 2

and obtain the following table

1

11

36

FUNCTIONALITY You have been

93

constructing curves by locating points from

a table and drawing a smooth curve through them

now see

method

that this

of plotting a function

is

you should

;

based on the

assumption that the given expression is a continuous function of In this case a small change in x makes' a small change in the given function hence if all values of x were taken, there would

x.

;

be a continuous succession of points forming a smooth curve.

In Fig. 52 imagine a perpendicular to the cc-axis drawn to = — 3. The length of this perpendicular is the value of the function the curve from x

x

for

=—

3.

Now

imagine

40

move

the perpendicular to

= + 4,

and you have a mental picture to the right to x

of the function varying con-

tinuously in value from to

+

11, then to

+

finally to

—

16,

— 41 and

36.

For certain intervals of values of x the function is greater than zero, and for certain intervals

nite

it

is

less

For certain defivalues of x the function

than

zero.

has the value zero.

The value

zero in the intervals

from x

of the function

=—2

to

x

=

is

greater than

.4 (about),

and from

= 2.9 (about) to x = + e». The function is less than zero from =— oo to = — 2, and from x = .4 (about) to x = 2.9 (about). The function has the value zero for x = — 2 and x = .4 (about).

x x

a;

Maximum and minimum

65.

—3

to

—

1, 2a;

As x

increases

+ 11

to

increases

—

16.

from

8

— 3x — 12* + 4 from — 1 to + 2, 2

As

—

a:

values. increases

As x increases from from

— 41 to + 11.

the function decreases from

from + 2 to +4, the function + 36. We observe that as the variable

increases

16 to

x increases continuously, the value of the function

may

either

APPLIED MATHEMATICS

94

At any

increase or decrease.

point where the function stops

increasing and begins to decrease,

value or to be a maximum. or

it is

said to have a

this case it occurs

maximum

when x

= — 1,

+ 11.

when

the function has the value

When

the function stops decreasing and begins to increase,

it is it

In

said to have a

occurs

when x

minimum

= 2, or when

value or to be a minimum. Here the function has the value

— 16.

In other words, a function is a maximum when its value is greater than the values immediately preceding and following.

In the same way a function is a minimum when its value is than the values immediately preceding and following. The point on the curve at which there is a maximum or

less

minimum

value of the function

is

called a turning point.

To investigate functional variation and get an idea of regional increase and decrease, and maximum and minimum 66.

values.

to give the shape of the curve. The and decrease are then readily noted. To check

Plot enough points

regions of increase

an apparent maximum

minimum

or

value of the function,

cal-

culate values of the function for points close together in the im-

mediate neighborhood and on both sides of the apparent value. That value of the function which is either greater or less than all those

which immediately precede or follow

is

the value desired.

PROBLEMS 1.

A

line

10

in.

long

is

divided into two segments which (a) Express

are taken as the base and altitude of a rectangle,

the area of the rectangle as a function of one of the segments. (b) Plot this function, (c) Discuss the increase and decrease of

area as the length of one segment changes from (d) What length of segment gives a maximum area ? is

the

maximum

Suggestion.

Let

area ?

(/) Is there a

minimum

x = one segment. — x = other segment. x (10 — x) = area. 10

to 10 in. (e)

area ?

What

FUNCTIONALITY

95

Express the sum of a variable number and its reciprocal number. Plot the function and investigate

2.

as a function of the

for regional changes.

of a

number and

An

3.

its

What

is

the

minimum

sum

value of the

reciprocal ?

open-top tank with a square base

contain 32 cu.

What

ft.

require the smallest

is to be built to should be the dimensions in order to

amount

Let

Suggestion.

of steel plate for construction ?

=

%

a side of the base.

—32

— = depth

Then

x2 x2

Plot the function x 2 H

=

+

of the tank.

surface of the tank.

x 128

and determine

a;

for the

minimum value.

x

Express the area of a variable rectangle inscribed in a whose radius is 4 in., as a function of the base. What are the dimensions of the rectangle of greatest possible area ? 4.

circle

Make

Suggestion.

note

how

and rectangle and

a drawing of the circle

the area changes as the base of the rectangle increases

from

to 8 in.

circle.

Why ?

Let

Then x

A

diagonal of the rectangle

is

a diameter of the

x = base of the rectangle. V64 — x2 = altitude of the rectangle. V64 — x2 = area of the rectangle.

Plot this function and determine the value of x that

makes

it

a

maximum. 5.

24

Show

in. is

What

6.

that the largest rectangle having a perimeter of

a square. are the dimensions of the greatest rectangle in-

scribed in a right triangle

8

whose base

is

12

in.

and

altitude

in. ?

Erom

7.

ber

is

the cube of a variable

function

number

six times the

num-

What value of the variable would make this a minimum ? Discuss the functional variation in full.

subtracted.

APPLIED MATHEMATICS

96

8. Prom a variable number its logarithm What value of the variable number would make a minimum ? 9. Two towns A and B (Pig. 53)

subtracted.

is

this difference

and 4 mi. respectively from

are 3

the shore of a lake CD.

If

CD

is

a straight line 7 mi. long, where

must a pumping to supply the

station

P

be built

towns with water with

the least amount of pipe ? If

10.

t

number of tons of and v represents the speed

represents the

steamer on a

trip,

hour, the following relation holds

:

t

=

.3

+

coal used

.001

The

11.

cost of an article

different prices

price

minimum

cost of a 1000-mi. trip a

is

is

Selling price in dollars articles sold

What

speed

?

35 cents. If the number sold

at

given by the following table, find the selling

which would probably give the greatest

Number

Other ex-

v*.

penses are represented by one ton of coal per hour.

would make the

by a

of the boat per

.

profit.

.50

.60

.75

.90

1.00

1.10

3600

3100

2640

2080

1300

700

Suggestion. First from the given table plot a curve to show the probable number sold at prices from 50 cents to $1.10. Then on the same axes with different vertical units plot the curve to show

the profits at the various prices.

number

To determine closely

Profit

=

(selling price

it

cost) x

the turning point of the second curve somewhat

will be necessary to locate intermediate points

the selling price at 80 cents and 85 cents. sold at these prices 12.

—

sold.

may

be found from the

The number first

;

e.g. for

probably

curve.

Devise other problems, in maxima and minima and

solve them.

CHAPTER IX EXERCISES FOR ALGEBRAIC SOLUTION IN PLANE GEOMETRY During the year given to plane geometry these exercises

67.

not only serve as a review of algebra, but they should also

develop in the pupils an ability to attack successfully

The

geometrical problems from the algebraic side.

many

figures for

exercises should be carefully drawn with ruler, comand protractor, and the drawing should check the algebraic work. Later the figures may be sketched. The numbers and letters should be put on the given and required parts in the drawing, and the equations set up from the figures. Bepresent lines, angles, and areas by a single small letter. Check all results.

the

first

es,

Complementary and Supplementary Angles 1.

20°

;

Find two complementary angles whose difference (d) x". (b) 52° (c) 5° 8' 10"

3.

4.

Find the angle that ;

(b)

7 times itself

;

.

is

(c)

—

:

30) are complementary

the complement of (a) 8 times

3 times

How many degrees are there

which are in the ratio (a) 1 2 5.

(a)

;

;

x/2 and x/3 (x + 40 and x angles. Find x and the angles. 2.

itself

is

? (6)

itself

in the

4 5 :

;

(d)

n times

itself.

complementary angles

? (c) 3.5

:

6.5

Find the value of two supplementary angles

?(d)m:n? if

one

is

9

(15) times as large as the other. 6.

How many

degrees are there in an angle that

plement of (a) 4 times (d)

n times

itself ? (b)

itself ?

97

7 times itself ?

(c)

is

the sup-

J of itself ?

APPLIED MATHEMATICS

98,

7. Of two supplementary adjacent angles, one lacks 7° of being 10 times as large as the other. How many degrees in

each

?

8.

If 10° (7°) be added to one of two supplementary angles

and 20° (8°) ratio 9.

2

:

5 (3

2

:

10.

Find the

4).

If 6° (5°) be taken

and added is

to the other, the resulting angles will be in the :

7 (13

to the other,

What

5).

:

To one

angles.

from one of two supplementary angles the ratio of the two angles thus found

are the angles ?

of two supplementary angles add 11° (9°) and

from the other subtract 16° will be to each other as 3 11.

How many

:

The two angles thus

(5°).

4(5

:

12).

its

? (b)

f of

its

complement

?

Find the angle whose supplement and complement added

together 13.

obtained

angles.

degrees are there in an angle whose supple-

ment is (a) 5 times its complement (c) n times its complement ? 12.

Find the

make 112° (208°).

If 3(8) times the

complement of an angle be taken from is 10° (76°). Find the angle.

supplement, the remainder 14.

an angle added to 5 times its supplement complement (supplement), what is the

If 3 times

equals 20 times

its

angle ? 15.

The angles formed by one

the ratio 7

:

11 (3

:

8).

How many

line

meeting another are in

degrees in each

?

16. Construct a graph to show the complement of any angle. (Take a large square each way equal 10°. Locate a few points and draw a x = 10, y = 80 x = 40, y = 50 x = 90, y = straight line through them.) What is the equation of this line ? :

;

;

;

17. On the same sheet of squared paper construct a graph show the supplement of any angle. What is the equation

to

of

the straight line ? 18. On the same sheet of squared paper as in the last two problems draw a straight line from (x = 0, y = 0) to (x = 80°,

EXERCISES FOR ALGEBRAIC SOLUTION y

=

Eead

160°).

What

this line.

99

a few pairs of angles given by points on

off

the equation of this line ?

is

mark the points that answer the

On

question, If one of

mentary (supplementary) angles

twice the other,

is

this line

two comple-

how many

degrees in each ? .

19.

Find two complementary angles such that the sum of

twice one and 3 times the other

is

210°.

Solve graphically.

20. Two complementary angles are in the ratio 2 3(7 Find the number of degrees in each. Solve graphically. :

21.

Three angles make up

The

point.

all

8).

:

the angular magnitude about a and second is 10° (20°), and

difference of the first

of the second

and third

How many degrees in each

100° (2°).

is

angle ? 22.

The sum

second

is

of four angles about a point

3 times the

of the first

first,

the third

is

The sum

360°.

10° greater than the

is

and second, and the fourth

is

twice the

first.

Find

the angles. 23.

Of the angles formed by two

5 (3 J) times another.

What

intersecting lines, one

is

are the angles ?

Parallel Lines 24.

Two

terior angle

by a transversal making one ex3 (5f ) times the other exterior angle on the same

parallels are cut

side of the transversal. 25.

If

two

Find

parallels are cut

all

the angles.

by a transversal making two ad-

jacent angles differ by 20° (36° 20'), find all the angles. 26.

If a transversal of

two

parallels

makes the sum of 5

(4)

times one interior angle and 2 (3) times the other interior angle on the same side of the transversal equal to 420° (625°), find all the angles. 27. The sum of one pair of alternate-interior angles formed by a transversal of two parallels is 8 (6£) times the sum of the other pair. Find all the angles.

:

APPLIED MATHEMATICS

100

Triangles 28. Of the angles of a triangle the second is twice the first, and the third is 3 times the second. How many degrees in

each angle ?

Find the angles of a triangle ABC, given (a) A 3 times and B 4 times as large as C. (b) A 3 times as large as C and B £ of C. (c) A ii and B 25° smaller than C. (d) A:B: C = 2 3 4(3 5 10).

29.

:

In a triangle

30.

C

J of angle

is

A

.

ABC

:

angle

:

A

is

Find the three

:

6 times angle B, and angle

angles.

Find the angles of the triangle ABC when B is 18° less than 4 times C.

31.

A

is

43° more

than § of B, and

The sum

32.

of the first

and second angles of a

triangle

is

twice the third angle, and the third angle added to 3 times the second equals 140° less the third angle. Find the three angles.

In a triangle the sum of twice the first angle, 3 times the is 320° (400°); and the sum of the first, twice the second, and 3 times the third is 440° (310°). Find the 33.

second, and the third

angles. 34.

sum

A and 35.

A lacks 106° of being equal to the lacks 10° of being equal to the sum of

In a triangle ABC,

of

B and

B.

The

C,

and C

Find the angles. vertical angle of

an

isosceles triangle is 68°.

Find

25° (47°).

Find

the base angles. 36.

One base angle

of an isosceles triangle

is

the vertical angle. 37.

Find the angles of an

isosceles triangle if a base angle

is

4(5) times the vertical angle. 38.

In an isosceles triangle the vertical angle

larger than a base angle.

Find the

is

36° (75°)

angles.

39. In an isosceles triangle 5 times a base angle added to 3 times the vertical angle equals 490° (530°). Find the angles.

EXERCISES FOR ALGEBRAIC SOLUTION Find the angles of an

40.

isosceles triangle in

101

which the ex-

terior angle at the base is 95° (140°). 41. The angle at the vertex of an isosceles triangle is ^-(^) of the exterior angle at the vertex. Find the angles of the triangle.

A

42.

base angle of an isosceles triangle

Find the angles of the

vertical angle.

What are

43.

is

12 (n) times the

triangle.

the angles of an isosceles triangle in which the

vertical angle is 12°

more than £(J) of the Sum of the base

angles ?

Construct a graph to show the change in the vertical

44.

angle y of an isosceles triangle as a base angle x increases from 0° to 90°.

The

45.

vertical angle of

an

isosceles triangle lacks 8° (20°)

Find

of being T73-(-9) of a right angle.

The

46.

all

the angles.

acute angles of a right triangle are x and 2 x (3 y

and 5 y). Find them.

The

47.

18° (37°).

difference of the acute angles of a right triangle

is

Find them.

48. If the acute angles of a right triangle are in the ratio (a) 2

:

3, (b)

7

:

8, (c)

m

:

n, find

the angles.

49. In a right triangle the sum of twice one acute angle and 3 times the other is 211° (192°). Find the angles.

Polygons 50.

How many

sides has a

polygon the sum of whose

inte-

rior angles is 720° (2340°)? 51.

An

interior angle of a regular polygon

How many side's has the polygon ? 52. How many sides has a polygon rior angles equals 53.

2 (12) times the

How many sides has

angles exceeds the

sum

sum

the

sum

is

165° (160°).

of

whose

inte-

of the exterior angles ?

a polygon the

sum

of the exterior angles

of whose interior by 1080° (2700°) ?

APPLIED MATHEMATICS

102

54. Construct a graph to show the sum of the angles of a polygon as the number of sides increases from 3 to 12.

Construct a graph to show the number of degrees in

55.

each angle of a regular polygon of n sides for values of n from 3 to 36. 56. If the number of sides of a regular polygon be increased by 2(3), each of its interior angles is increased by 15° (10°). How many sides has the polygon ?

57. By how many must the number of sides of a regular polygon of 12(15) sides be increased in order that each interior angle may be increased 18° (6°) ?

By how many must

58.

the

number of

polygon of 8 (20) sides be increased diminished 5° (6°)?

if

sides of a regular each exterior angle is

Construct a curve to show the number of degrees in an

59.

exterior angle of a regular polygon as the

number of

sides

increases from 3 to 18.

The perimeter

60.

and the sides are

The perimeter

61.

3

:

of a triangle

as 1

1 (15

:

and

4)

The sum

62. ft.

e is

4

;

4(2 5 :

:

176(50.4)

is

7).

Find the

ft.

in length

sides.

of a triangle bears to one side the ratio

is

the third side

4 1 (5 :

:

2).

What

?

b, and c, of a triangle is than the sum of b and c, and twice more than the sum of a and b. Find each side.

twice a

ft.

:

to another side the ratio

part of the perimeter

35

3

:

of the three sides, a,

is

5

ft.

less

and base of an isosceles triangle are in what part of the perimeter is one of the

63. If the perimeter

the ratio 4

:

1 (5

:

2),

equal sides ? 64.

Find the perimeter of an

isosceles triangle if it is 4 (8£)

times the base, and one of the equal sides

is

4 (55)

ft.

longer

than the base.

In an isosceles right triangle the perpendicular from is 12 (30) cm. long. How long is the hypotenuse ? 65.

the vertex to the hypotenuse

EXERCISES FOR ALGEBRAIC SOLUTION

103

hypotenuse of an isosceles right triangle is 26 (8) in. the length of the perpendicular from the vertex to the hypotenuse ? 66. If the

long,

what

is

Parallelograms 67.

One angle

68.

An

of a parallelogram

Find

tive angle.

all

angle of a parallelogram

Find

other angles.

is

4 (9) times

is

3(2f) times one of the

all

70.

71.

:

1 (4

:

5),

the difference of

how many

in the

degrees in each angle ?

How many degrees in each angle of a parallelogram when

an angle exceeds 72.

if

two consecutive angles of a parallelogram are

If

17

consecu-

the angles.

69. Find the angles of a parallelogram two consecutive angles is 20° (90°).

ratio

its

the angles.

-J

The number

(£) of its consecutive angle

by 30° (56°) ?

of degrees in one angle of a parallelogram

equals J of the square of the number of degrees in the consecutive angle. Find all the angles. 73.

if two angles x and y of a quadsupplementary, the other two 'angles a and b are

Prove algebraically that

rilateral are

also supplementary. 74.

Find the

sides of a parallelogram if one side is § (|) of

another side and the perimeter 75.

One

other side 76.

and the perimeter

The sum

is

28 (84) cm. 4 (5)

is

36 (58)

in.

in.

longer than an-

Find the

of two adjacent sides of a rhomboid

the difference of those sides. is

is

side of a parallelogram

Find the sides

if

sides. is

$ (f ) of

the perimeter

18.3(82) cm. 77.

One angle

of a

rhombus

is 60°.

If 5 (2) times the perim-

eter exceeds the square of the shorter diagonal

by 19(13|),

find a side of the rhombus. 78. In a rhomboid two of whose sides are a and b, 3 times a exceeds twice b by 11, and the sum of twice a and 5 times b is 20. Find the perimeter.

APPLIED MATHEMATICS

104

79. In one of the triangles formed by the diagonals of a rhombus and one of the sides of the rhombus the two smaller angles are in the ratio 2 3(1 3). Find all the angles of the :

:

rhombus. 80. The perimeter of a parallelogram is 16(9.6), and the square of one side added to 4 (2) times an adjacent side equals

Find the sides of the parallelogram.

37(8.6). 81.

In a rhombus one of whose angles is 60° the shorter is 10 in. (5 ft. 6 in.). Find the perimeter.

diagonal 82.

Two

sides of a rectangle are x

the perimeter

is

60(40).

Find the

and x 1 (3 x and 7 x) and

sides.

Circles 83.

The circumference

of a circle

is

divided into three parts.

Find the number of degrees in each part if the second contains 3(6) times as many as the first part, and the third part contains 5 (7) times as 84.

In a

many

as the first part.

a diameter and a chord are drawn.

circle

The

4(5) in. longer than the chord and the diameter and chord together are 18 (20) in. long. How long is each ?

diameter

85.

chord.

is

There are 100° (x°) in one of the arcs subtended by a

How many

degrees are there in the other arc ?

86. In one of the arcs subtended by a chord there are 50° (120°) more than in the other arc. How many degrees

in each arc ? 87.

radius 88.

Find the side of a square inscribed in a is 30 (42.5) mm.

A

triangle

in a Circle.

The

Find the three 89.

In a

whose perimeter

first

side

is

is

36 (72)

mm.

circle

is

whose

inscribed

£ of the second and § of the third.

sides.

circle of radius

in length to the radius.

8(12)

How

far

in.

is drawn equal from the center ?

a chord

is it

EXERCISES FOE ALGEBRAIC SOLUTION

A circle containing 280 (308) sq.

90.

The

parts by radii.

the second part

is

ft. is

105

divided into three

and Find the

third part equals twice the second,

20

sq. ft. larger

than the

first.

area of each part.

A line

91.

1 (3.6) ft.

of the part outside,

chord

long intersects a circumference in two

If the part inside the circumference is twice the length

points.

how

long

is

the part which forms the

?

A

number of coins are placed in a row touching one and the length of the row is measured. 3 quarters, 2 nickels, and 5 dimes measure 204 mm. 1 quarter, 3 nickels, and 2 dimes measure 123 mm. and 1 quarter, 1 nickel, and 1 dime measure 63 mm. Find the diameter of each coin. Check. 92.

another,

;

;

A

93.

boy has 20 copper disks

;

part of them are 20

mm.

in

diameter and the rest are 30 mm. The sum of their diameters is 520 mm. How many of each kind has he ?

Two

94.

diameters are drawn in a

circle,

making

at the

center one of the supplementary adjacent angles 3 times the

How many

other.

A

95.

degrees in each angle ?

chord 6 (4)

A

96.

chord 16(4)

the center of a circle.

3 (1)

A

long

98.

center

in.

is

chord 8(12) is

in.

from the center of a

at a distance of 6(8) in. from the length of a chord which is

is

is

?

in.

long bisects at right angles a radius.

the radius ?

The radius is

4 (6)

circle.

long

What

from the center

in.

97.

How

long

in.

Find the radius of the

circle.

of a circle

a chord 8 (4)

in.

is

5 (3)

in.

How

far

from the

long ?

99. The radius of a circle is r. What is the length of a chord whose distance from the center is ^ (J) r ?

100. Find the length of the longest and shortest chords that can be drawn through a point 9 (6) in. from the center of a circle

whose radius

is

15 (8)

in.

APPLIED MATHEMATICS

106

The sum

of the longest and the shortest chords through from the center of a circle is 18 (64) in. Find the radius and the two chords. 101.

a point 3 (8)

in.

102. Construct a curve to

show the length of a chord

circle of radius 8 in. as the distance of the

center increases from

A

103.

legs are 6

to 8 in.

circle is circumscribed

and 8 (5 and 12)

The

104.

in.

about a right triangle whose


the point of tangency P, a distance

A

106.

in.

measured

PA

equal to

on the tangent. If the distance to the center of the circle is 10 (6) in., find the radius.

twice the radius

from

13 (5)

circle is

sides of the triangle.

From

105.

circle.

legs of a right triangle inscribed in a circle are

5 x and 12 x (x and 3 a;) and the radius of the

Find the

in a

chord from the

In a

is

circle of radius 8 (5) in.

opposite sides of the center. as the other.

123 (10)

in.,

off

sum

If the

One

is

two

parallel chords lie on

twice as far from the center

of the squares of the half chords

find the distance each chord

is

from the

is

center.

107. The perimeter of an inscribed isosceles trapezoid is 38 (88) in. One of the parallel sides is f (.7) of the other and one of the nonparallel sides is 9£ (30) in. shorter than the longest side of the trapezoid. Find each side. 108.

Two

circles

touch each other and their centers are

8 (a) in. apart. The diameter of one diameter of the other ? 109.

Two

circles are

12(5)

110.

18 (a)

Find the

The in.,

10 (d)

tangent externally.

in.

The

and the distance between

their radii is 8 (a) in. in.

is

What

is

the

difference of

their centers

is

radii.

between the centers of two circles is one half the sum of their radii. Find the

distance

which

is

radii.

111.

of

its

One angle

of an inscribed triangle

sides subtends

angles of the triangle.

is 35° (50°) and one an arc of 113° (150°). Find the other

EXERCISE FOR ALGEBRAIC SOLUTION 112.

The circumference

in the ratio 1

:

2

:

3(2 3 :

:

of a circle 5).

formed by the chords of the

A

is

107

divided into three arcs

Find the angles of the triangle

arc.

The sum .of the first and third angles is twice the second angle, and the difference of the first and second is 20°. How many degrees in each of 113.

triangle is inscribed in a circle.

the three arcs ? 114. Construct a

angle

graph to show the change in an inscribed by its sides increases from

as the arc intercepted

y,

to 180°. 115.

An isosceles triangle is inscribed in a circle. The number

of degrees in the arc upon which the vertical angle stands

is

8(3£) times the number of degrees in a base angle of the triangle.

Find the angles of the

triangle.

an inscribed quadrilateral subtend arcs of 82°, 99°, 67°, and x° respectively. Find each angle of the quadrilateral also each of the eight angles formed by a side and a diagonal. 116. Consecutive sides of

;

117.

How many

degrees in each angle of a quadrilateral

inscribed in a circle, if the sides subtend arcs

the ratio 118.

which are in

1:2:3:4(2:3:5: 6)?

A

right triangle

angle of the triangle

is

is

inscribed in a circle.

§ (f ) of the other,

If one acute

how many

degrees in

each of the three arcs ? 119.

A BCD is

anjnscribed trapezoid. If the angle

A

is

twice

angle C, find each angle. 120.

Two

The angle

chords

APC

is

AB and CD intersect within a circle 50°, arc DB is 40°, and arc AD is

at P. 160°.

Find the other arcs and angles.

Two chords AB and CD intersect within a circle at P. BD is twice are AC, and arc CB is twice arc DA. Angle DPA is twice angle APC. Find the arcs and angles. 122. The angle y is formed by two chords AB and CD intersecting in a circle, and the two intercepted arcs A C and DB are 121.

Are

APPLIED MATHEMATICS

108

90° and x° respectively. What is the equation connecting y and x ? Construct a graph to show the change in y as x to 90°increases from

From a

123.

point without a circle two secants are drawn,

making one the

sum

of the intercepted arcs 3(5) times the other. If of the other two arcs is 200° (300°), what is the angle

formed by the secants

The angle y The

124.

out a

What to

circle. is

show

formed by two secants intersecting with-

intercepted arcs are 90° and x° (x

< 90).

the equation connecting y and x ? Construct a graph to 90°. the change in y a,s x increases from

Two

125.

is

?

tangents drawn from an exterior point to a circle of 60° (80°). Find the two arcs. the points

make an angle of tangency

and

find the other

two angles

in the triangle thus

formed.

Through the ends of an arc of 45° (100°) tangents to the circle are drawn. Find the angle formed by the tangents. Find the other two angles in the triangle formed by ing 126.

the points of tangency. 127. Find the angle formed by two tangents to a circle drawn from a point at a distance from the center of the circle equal to the diameter.

128.

From

P, a point without a

circle,

PA

two tangents

and

PC are drawn. The arc AB equals 160° (100°). difference of the angles BPC and A is 10° (25°), find

PB, and a secant If the

the angles. 129.

From a

point without a circle of radius 4 (8)

in.

a

secant through the center and a tangent are drawn. If the angle formed by the secant and tangent is 30° (60°), find

the distance from the point to the center of the

circle,

and

the length of the tangent. 130.

a

In an equilateral triangle whose sides are 40 (60) mm.

circle is inscribed.


radius of the circumscribed circle.

circle.

Find the

:


109

Ratio Express the ratio of the following pairs of numbers in the simplest form 131.

(a) 168 and 252. (b) 387 and 602.

(h) 148 a 3 and 185 a; 4 (i) x 2 + 5x + 6 and x .

(d) 5% and 30J. (e) £f and |. (/) .125 and 3.75. Iff)

(b)

a a

and

a;

on the

lines

a and

Find the

b.

(c)

a

(rf)

a

= 4 cm., 5 = 12 cm. = 14 mm., 6 = 35 cm.

a sheet of squared paper let the bottom line be the left border line be the y-axis, and the side of a

and the

square each (0, 0)

2

:

= 5 in., & = 10 in. = 3^ in., 6 = 7 in.

On

133. as-axis

2

a;

.

132. Squares are constructed

(a)

2

W ,

4

ratio of the areas

2

2

a:

6a £cand30a «. 2

+ 3. + 2x — 15 and a + 5. + 3x + 2 and* + Ax + 3. (0 + 6a; + 5 and a + 8ai + 15. x + 2 +6a; + 8 an x + 3 + 7* + 12' 2 (j) x (Y) x 2

(c)'fand£.

and

way

What

y.

= 1.

Draw a straight line through the points Make a table of corresponding values of x the ratio of y to x ? What is the equation of

(8, 16). is

the line ?

The width y

of a field is to be made § of the length x. the equation connecting y and x ? Construct a graph to show the width of the field for a length from 10 to 100 rd. 134.

What

is

135. If the ratio of

What

the relation.

14 x

136. If

is

y to x

is

2

:

3,

construct a graph to

the equation of the straight line

— 9 y = 2 x — y, find the ratio oix:y.

show

?

Construct

the graph. 137.

What

138. If x

:

is

the ratio oix:y,

y=4

:

if 7a;

— 6y = 3x +
5, find the value of the ratio 2

:

Construct the graph. 139.

=

1:2.

Find the value of the

ratio 3 x 2

+ 2y

2 :

xy

+y

2 ,

if

x y :

':

APPLIED MATHEMATICS

110

Proportion 140. Test the correctness of the following proportions

W

_84

180

_42 ~ 90'

1.25

.

C

141.

203

~ 259

.26

a2

_48__96 (' 225 ~ 45' _87__111

^

}

(*'

120 24

+ 2 ab + b* _ a + b a?-b ~a-b' + 7a; + 10 = x + 2 +5 + 5)

2

(/)

2

(a;

a;

Find <e in the following proportions 2

32

18

'

2

a; '

_ ~

- W-

:

a-b

^25 = ^'

W^TT = ^-'

W

W

,

s

.

18'

x

(C) _£.

M

_

7N

#\

ffl

a3 «"

(/)

4.8~.24'

- 5 a -fi' * + 10 « + 25 _ ~9' s

x

What number can be added to 7, 12, 1, and 3 (5, 19, and 52) so that the resulting numbers will form a pro-

142. 16,

portion ?

Find the numbers proportional to 1, 2, 3, 4 (2, 5, 1, be added regularly to 5, 10, 15, 40 (11, 20, 8, 14) as to form a proportion. 143.

that

may

144.

The

(.4)

145.

so

line ing the mid-points of the nonparallel sides

of a trapezoid

|

3)

is

20 (42)

Find the bases

in. long.

if

one

is

of the other.

In a triangle

the side

AC

ABC

in the ratio 3

the line :

4 (5

:

PQ

9).

If

parallel to

AB =

20

BC

divides

(9.8) in., find

the two segments of AB. 146.

The sum

is 45 (63) in. A from the vertex segments Find the two sides.

of the two sides of a triangle

line parallel to the third side cuts off

10 and 8 (4 and 20)

A line 100 (6) ft. long is divided into parts in the 2 3 4 (2 3 7). Find each part.

147.

1

:

in. long.

:

:

:

:

ratio

EXERCISES FOR ALGEBRAIC SOLUTION 148.

111

Three lines are in the ratio 2 3 4 (2 1 6) and their is 30 (24). Find the length of each line. :

:

:

:

fourth proportional 149.

The sum

third side, 18

two

sides.

150.

of

sides of a triangle

long,

is

is

20

(5) in.

The

a third proportional to the other

Find them.

One

side of a triangle

and the third side a

two

(4J-) in.

is

5

in.

2

is

in.

longer than the

longer than the

first.

first side,

If one side

is

mean proportional between the other two, find the three sides. 151. The three sides of a triangle are x, y, and 3. The cor-

responding sides of a similar triangle are 10, 20, and 15. Find

x and

y.

The sum of the three sides of a triangle, x, y, and a, is and the corresponding sides of a similar triangle are x + 3, y + 7, and z + 5. Find the sides of each triangle. 153. The three sides of a triangle are 3x, 6x, and 8* (x, x + 1, x + 2), and the corresponding sides of a similar triangle are 3 a; 2 6 a; 2 and 8 a; 2 (a; 2 x 2 + x, and a; 2 + 2 a;). If the sum of the perimeters of the two triangles is 102 (75), find the 152.

15,

,

,

,

sides of each triangle. 154. The sides of a triangle are 5, 8, 12 (12, 16, 20) in. Find the segments of each side made by the bisector of the opposite angle.

The sum

155.

of

two

sides of a triangle

is

24

in.,

and the

bisector of the included angle divides the third side into parts

4 and 8

in. long.

Find the three

sides.

In a triangle ABC, AB = 12 and BC = 36. From a point on AB at a distance x from A a line y is drawn to AC parallel to the base. Construct a graph to show the length of y as x increases from to 12. 156.

Right Triangles 157.

angle

The hypotenuse

is 30°.

Find

of a right triangle

(a) tile other

two

sides

;

is

(b)

8

in.

and one

the perpendic-

ular from the vertex of the right angle to the hypotenuse

the segments of the hypotenuse.

(c) ;

APPLIED MATHEMATICS

112 158.

One

leg of a right triangle is 2 (3)

other and the hypotenuse leg.

Find the three

159.

The

Find

ft.

is

4 (7)

ft.

ft.

longer than the

longer than the shorter

sides.

12 and 16 (5 and 12) the perpendicular from the

legs of a right triangle are

(a) the

hypotenuse;

(6)

vertex of the right angle to the hypotenuse

;

the segments

(c)

of the hypotenuse. 160.

The perpendicular from the vertex

of a right triangle to the hypotenuse

hypotenuse

is

26 (6.25)

in. long.

is

of the right angle

12 (3)

in.

long and the

Find the other two

161. If the legs of a right triangle are

a and

b,

sides.

find the per-

pendicular from the vertex of the right angle to the hypotenuse,

and the segments of the hypotenuse. 162.

One side of a right triangle is 4. Construct a curve to show

the length of the hypotenuse as the other side increases from to 16.

the

(Let the bottom line be the x-axis, the left border line be and the side of a large square each way 1. Take the

=

t/-axis,

and locate the points of the curve with comes. Check a few of the points by computation.) side 4 on the vertical axis

Chords, Tangents, Secants 163.

The segments

7 and 9 (15 and 13) is

made by another chord are and one segment of the latter chord

of a chord

in.,

in. What is the other segment ? Two chords intersect, making the segments of one chord

3 (10) 164.

2 and 12(4 and 8) in., and one segment of the other chord 2 (14) in. longer than the other segment. Find the two chords. 165.

One

of

two intersecting chords

is

14(17) in. long, and is 45 (60). Find

the product of the segments of the other chord

the segments of the 166.

Two

first

chord.

secants intersect without a circle.

The

external

20 (2) in. and the internal segment is 5 (4) in. If the external segment of the other secant is 10 (3) in., find segment of one

is

the length of the internal segment.


113

167. From a point without a circle two secants are drawn whose external segments are 5 and 6 (6 and 8) in. The internal segment of the former is 13 (16) in. What is the internal segment of the latter? What is the length of the tangent from the same point ? 168. Two secants from a point without a circle are 24 in. and 22 in. long. If the external segment of the lesser is 5 in., what is the external segment of the greater.? What is the length of the tangent from the same point ?

A

169.

tangent and a secant are drawn to a circle from an

external point.

The external and

internal segments of the

secant are respectively 2 (3) in. and 1 (4) in. shorter than the tangent. What is the length of the tangent ?

From a

170.

point on the tangent of a circle 6(15)

ment

is

in.

from

drawn whose internal seg2(3) times the external segment. Find the length of

the point of tangency a secant

is

the secant. 171.

A

tangent intersects a secant which is drawn through The length of the tangent is 4(t) in.,


and the length of the external segment of the secant in. Find the radius of the circle and the secant.

is

2(s)

17 in. a point P is taken on the from the center. What is the product of the segments of chords through P? Denoting the segments by x and y, what is the equation that connects x and y ? In this equation give values to x and make a table of values of x and y. Construct a curve to show the change of y as x increases from 2 to 32 in. 172.

In a

diameter 15

173.

circle of radius

in.

From a

point on the circumference of a circle of 9

diameter a tangent 6

in.

long

is

drawn.

From

in.

the end of the

If y is the external and x the insegment of the secant, what is the equation connecting x and y? Construct a curve to show the length of y as x increases from to 9 in. and then decreases to 0.

tangent secants are drawn. ternal

APPLIED MATHEMATICS

114

Akea of Polygons 174.

The base

the area 175. is

7

is

of a triangle

90 (75)

The area

176.

5(3) times the altitude and

is

Find the base and

altitude.

is 130(42) sq. in. and the altitude more) than the base. Find these dimensions.

of a triangle

in. less (5 in.

12 (23)

sq. in.

The sum of the base and altitude of a triangle is and the area is 16 (45) sq. in. Find the base and

in.

altitude.

177. Find the area of a right triangle whose base is 20(32) and the sum of whose hypotenuse and other side is 40 (50). 178.

The altitude and

its sides

179.

The

altitude of a triangle

If the altitude is

is

increased 52 sq. 180. If the

than one

of an equilateral triangle

is

12 (A)

ft.

Find

area.

leg,

is

16

than the base. in., the area

in. less

and the base 12 Find the base and altitude.

increased 3 in.

in.

hypotenuse of a right triangle is 1 (8) in. longer and 8 (9) in. longer than the other leg, what is

the area of the triangle ?

16

V3 (60) sq. in.,

and h the

altitude of an

an equilateral triangle and a side.

181. If the area of

find the altitude

a denotes the area,

182. If

s

a

side,

is

equilateral triangle, express each in of the others. 183. If a rectangle is 7(8)

contains 170 (209) sq. 184.

length 185.

times

ft.,

The perimeter 2 (5) times

is

of a rectangle

its

Find

width.

is

wide and

is

72 (132)

ft.

and

its

its area.

rectangle whose length is 8 (5) ft. greater than 3 (4) width contains 115 (3750) sq. ft. Find its dimensions.

The area of a rectangle show the altitude as the base

187.

it

A

its

186.

to

longer than

ft.

find its dimensions.

The

side of one square

another square, and

its

of the second square.

area

What

is is

is

36

sq. ft.

Construct a curve

increases from 1 to 36

is

ft.

3 (4) times as long as that of sq. yd. greater than that

72 (90)

the side of each square ?


.

115

188. One side of a square is 3 (6) yd. less than 2 (3) times the side of a second square, and the difference in area of the squares is 45 (756) sq. yd. Find the area of each square.

One side of a rectangle is 10 (6) ft. and the other side 2(1) ft. longer than the side of a given square. The area of the rectangle exceeds that of the square by 80 (174) sq. ft. 189.

is

Find the side and area of the square. 190. The floor of a rectangular room contains 180 (240) sq. ft., and the length of the molding around the room is 56(62) ft. Find the length and width of the room.

A

191. it is

picture including the frame

192 (480) 192.

by 16 its

193.

sq. in.

What

is

10(9)

which

in.

is

288(138)

is

sq. in. ?

Around a square garden a path 2

ft. wide is made. If taken for the path, find a side of the garden.

Around a garden 100

ft.

by 120

ft.

a

man

make, a path which shall occupy 57„ (£) of the area. must the path be made ? 195.

longer than

3(6) in. wide, are the dimensions of the picture ? is

The dimensions of a picture inside the frame are 12 in. by 12 in.). What is the width of the frame if

sq. ft. are

194.

of the frame,

in. (5 in.

area

376

The area

wide.

A

wishes to

How

wide

rectangular building having a perimeter of 140

ft.

by a fence whose distance from the building is J the width of the building. If the area between the fence and building is 1800 sq. ft., find how far the fence is from the building. is

inclosed

196.

An

open-top box

is

made from a square

piece of tin

cutting out a 5 (2)-in. square from each corner and turning

How large

the sides.

180 (242) 197.

is

the original square

if

by up

the box contains

cu. in.?

An

open-top box

is

formed by cutting out a

l(3)-in.

square from each corner of a rectangular piece of tin 2 (3) times as long as it is wide, and turning up the sides. If the total surface of the box

the piece of

tin.

is

284 (936)

sq. in., find the

dimensions of

.

APPLIED MATHEMATICS

116

make an open-top box from a piece of 30 (24) (15) in. sq., by cutting out equal squares from each corner and turning up the strips. What should be the length of a side of the squares cut out to give a box of the greatest possible volume ? 198. It is desired to

tin

Suggestion.

y

=

x

If

=

side of square cut out,

volume

of the

box

=

- 2 xf.

x (30

Make a table of values of y, giving x the values 1, 2, 3 Locate the points and draw a smooth curve through them. The turning point of the curve will show the value of x for the •

•

•

greatest volume. 199.

by 36

From

a rectangular piece of tin 12 desired to

in.) it is

in.

by 24

make an open-top box

in.

(16

in.

of the largest

by cutting out equal squares from the corners and turning up the strips. What should be the length of a side possible volume,

of the squares 200.

The

10(455)

?

altitude of a trapezoid

sq. in.,

Find the

and the

is

5 (14)

in.,

difference of the bases

the area

is

2 (11)

in.

is

bases.

The area

of a trapezoid is 90 (495) sq. ft., the line ing the mid-points of the nonparallel sides is 6 (45) ft., and the 201.

difference of the bases

is

202. In a trapezoid b

a the

area.

203.

from

Find each

The base to

20

in.

6 (12)

and

b'

ft.

Find the bases and

altitude.

are the bases, h the altitude, and

in of the other.

of a triangle

is

12

in.

and the

altitude increases

Construct a graph to show the increase in

area of the triangle. 204. The base and altitude of a triangle increase uniformly, and the altitude is always twice the base. Construct a curve to show the change in the area of the triangle as the base increases from to 10 ft. 205.

The base and

respectively.

What

parallel to the base

is

altitude of a triangle are

24

in.

and 9

in.

the area of the triangle formed by a line

and 6 (8)

(x) in.

from the vertex

?


117

whose base is 12 in. and altitude is 16 in, drawn parallel to the base and at a distance x from the If y = the area of the triangle cut off from the vertex, the equation connecting x and y ? Construct a curve

206. In a triangle

a line

is

vertex.

what is show the area of the to 16 in.

to

triangle cut off as x increases

from

207. The altitude of a triangle is 2 (3) times its base. Through the mid-point of the altitude a line is drawn parallel

If the area of the triangle cut off

to the base.

find the base

and

The sum

208.

is

36(5)

of

the

and the

of

areas

two similar

triangles

The difference of the areas of two squares and a side of one is 3(14) ft. longer than a Find a side of each square.

209. sq. ft.,

other.

The sum

210.

and a

of the areas of

two squares

is

is

triangle.

39 (324)

side of the

13(221)

sq.

ft.,

1 (9) ft. shorter than a side of the Find a side of each square.

side of one square

other.

is

sides of one are 2 (2 J) times the cor-

240(290) responding sides of the other. Find the area of each sq. in.,

sq. in.,

altitude of the given triangle.

is

A

211. side of one square is 5(2) in. longer than a side of another square, and the areas of the squares are in the ratio

4

:

1 (16

:

What

9).

is

a side of each square

212. Construct a curve to sides increase

from

Circles

show the area

of a square as

its

to 13 in.

and Inscribed Polygons

213. Construct a curve to

radius increases from

?

show the area

to 16 in.

of a circle as

(Locate points for r

its

= 0, 2, 4,

...,16.)

214.

The

radius of a circle

is

5(8)

(r) ft.

Find a side and

the area of the inscribed square. 215.

What

whose area

is

is

the radius of the circle inscribed in a square

1600(5000) (a)

sq. ft. ?

APPLIED MATHEMATICS

118 216.

6 (12)

An

equilateral triangle

Find a

(r) in.

217. (s) in.

The

side of

is

inscribed in a circle of radius

side, the altitude,

and area of the

an inscribed equilateral triangle


is

triangle.

9 (1.732)

circle.

218. The sum of the side of an inscribed equilateral triangle and the radius of the circle is 5 + 5 VE (10.928) (18) in. What is the length of a side and the radius ?

219. (a) sq.

The area of a regular

inscribed hexagon


ft.

An

is

V3 (17.32)

24

circle.

and a regular hexagon are inFind the radius of the circle if the sum of the areas of the triangle and hexagon is 9 V3(l8V3) (389.7) 220.


scribed in a circle.

sq. in.

221. The sum of the perimeters of two regular pentagons is 100 (225) ft., and their areas are in the ratio 1 9 (25 16). Find a side of each pentagon. :

222.

The

difference of the perimeters of

and

ft.,

223.

The sum

20 7T (176)

What 224.

ft.,

of

the

and the

circumferences

difference of

The radius

113$. (31.416)

225.

of

:

:

two

their radii

is

circles

2 (14)

is ft.

are the radii? of one circle

What

ft.

is

Find the

is

6 (1)

sum

radius of another circle, and the is

two regular octagons 4(9 25). Find

their areas are in the ratio 1

40(80) a side of each octagon. is

:

ft.

longer than the

of their circumferences

radii.

the radius of a circle whose area equals the in. ? (b) 3.3 and 5.6 cm. ?

area of two circles of radii (a) 3 and 4 (c) 6.5

226.

sum

and

7.2 cm. ? (d) r

What

is

of (a) 3, (b)

227.

What

by increasing

is

its

and nr

?

the radius of a circle whose area equals the 6, (c)

n equal

circles ?

the radius of a circle that radius 1 (3)

ft.

?

is

doubled in area

EXERCISES FOR ALGEBRAIC SOLUTION 228.

A

119

square and a circle have the same perimeter. Find

the ratio of their areas. 229. If a square

and a

circle

have the same

area,

what

is

the ratio of their perimeters ? 230. If a circle

perimeter,

what

is

231. Construct

and an

equilateral triangle have the

same

the ratio of their areas ?

on the same axes curves to show the change in

area of a circle and the inscribed regular hexagon, square, and equilateral triangle, as the radius increases 232. sq.

ft.

from

ft.,

10

in.

The area between two concentric circles is 20 77- (286) and the difference of the radii is 2 (7) ft. Find the radii. between two concentric circles and the radius of the inner circle is 2 (5)

233. If the area sq.

to

is ft.,

96 ir (50) find the

radius of the larger circle. 234.

In a

circle of radius

12 (f)

in. it is

desired to draw a

concentric circle which shall bisect the area of the given circle.

Find

its

radius.

235. The area of a circle of radius 8 (r) in. is to be divided by a concentric circle so that the area of the ring shall be a mean proportional between the area of the given circle and of the inner circle. Find the radius.

CHAPTER X COMMON LOGARITHMS 68. Definitions. 10.

Thus 2

=

Numbers have been reduced

lO 03010 3 ,

=

10°-

4m 125 ,

=

10 2 0969

to powers of

-

.

These exponents are called logarithms. The integral part of a logarithm, called the characteristic, can be determined easily

and

is

not given in a table of logarithms

;

the decimal part,

called the mantissa, is always taken from the table. 69. Approximate numbers. In ordinary shop practice and in much engineering work measurements are made usually to three or four figures. Thus in making a rough estimate the sides of a building lot may be measured to the nearest foot; the length of a belt may be measured to the nearest quarter of an inch an angle may be measured to the nearest tenth of a degree. If the diameter of a pulley is measured and said to be 12.3 in., the meaning is that the diameter lies between 12.25 in. and 12.35 in., that is, the third figure is doubtful. In ordinary computations, where numbers with only three or ;

four figures are involved, a four-place table of logarithms used.

The logarithms

are not exact

;

is

they are approximate

numbers in which the fourth

figure is doubtful.

results should not be carried

beyond four

Hence

the

figures.

The mantissa. To find the mantissa of the logarithm number from 1 to 999, e.g. 352, we look in the first column of the table at the left for the first two figures, 35, and in the column headed 2 we find the mantissa of 352, namely .5465. The mantissa of 745 is .8722. (Let the class read the mantissas of numbers from the table till all can find the mantissa of any number quickly.) 70.

of a

120

:

COMMON LOGARITHMS 71.

The

characteristic.

teristic is readily s

10 =1000,.

The method

121

of finding the charac-

obtained from the following

table,

APPLIED MATHEMATICS

122 log 216

= 2.3345,

Dividing sides of the

tion

by

log 2

both

10 2 8846 10

equa-

JoTIiS

i.e.

10,

= 0.3010,

i.e.

10 08846

=2.16

=10

10 9.8846 - io

=

= 2. = 100

102.8oio

=200

-

2

= 1.3345.

log 2.16

= 0.3345.

log .216

= 9.3345. -10.

.

216

8010

10

.-.log21.6

1()

10

10

sides of the equa-

by

_

10

Multiplying both tion

= 216. = 10 = 21.6

-

.

• .

.

.\ log

200 = 2.3010.

100,

Hence

it is

seen that moving the decimal point any number

is multiplying or dividing by some and this affects only the characteristic. The mantissas of numbers having one, two, or three figures are taken directly from the table. The mantissas of four-figure numbers are easily found. Find the logarithm of 1836. The mantissa of 1836 is the same as the mantissa of 183.6, since moving the decimal point does not change the mantissa. The mantissa of 183.6 lies between the mantissas of 183 and 184 and it is .6 of the way from the mantissa of 183 to the mantissa of 184.

of places to the right or left

integral

power of

10,

;

Mantissa of 184

= 2648 - 2625 = 23. 23 x .6 = 14. 2625 + 14 = 2639. log 1836 = 3.2639.

- mantissa

of 183

.-.

Find log 49.23. Mantissa of 493

- mantissa of

.-.

= 6928 - 6920 = 8. 8 x .3 = 2. 6920 + 2 = 6922. log 49.23 = 1.6922. 492

COMMON LOGARITHMS To find the

logarithm, of a number.

(mentally) after the third figure.

from

123

Place the decimal point

man-

Subtract the next lower

Multiply the difference by the fourth the number regarded as tenths, disregarding a fraction figure of less than one half and calling one half or more one ; add the prodtissa

the next higher.

uct to the next lower mantissa.

Write the proper characteristic.

(Let the class find the logarithms of

work should be done mentally

;

it

many numbers. The

can be done easily and quickly

with practice.) 73.

To

find

a number from its logarithm. Given log b = Looking in the table of mantissas, b.

1.5927, required to find it

is

seen that 5927

lies

between 5922 and 5933; the

cor-

responding numbers are 391 and 392. Hence the number corresponding to 5927 plus a fraction.

lies

To

between 391 and 392 that is, it add a zero to the ;

find the fraction,

ence of the given mantissa and the smaller, and divide difference of the next larger

it

is

391

differ-

by the

and next smaller mantissas. 5922 5927 5933

391 391.5

392

11)50(5 Since a difference of 11 in the mantissas makes a difference of

makes a difference of T5T in Hence the mantissa 5927 gives the number But the characteristic 1 shows that there are

1 in the numbers, a difference of 5

the numbers.

391^ =

391.5.

two integral Given

figures in the

logm = 0.9145, log n = 8.8132 -

To find a number from tissa lies

number. Therefore

its

m= 10,

n

8.213..

= .06504.

logarithm.

between two mantissas in the

b == 39.15.

When

the given

man-

table, divide the differ-

ence of these mantissas into the difference of the smaller mantissa

and the given mantissa,

to

one decimal figure.

Add this decimal

:

APPLIED MATHEMATICS

124

to the number corresponding to the smaller mantissa and place the decimal point in the position indicated by the

figure

characteristic.

number from its logarithm should be it can be done easily and quickly.)

(All the work in finding a

done mentally

with practice

The use of logarithms in computation.

74.

are exponents I.

;

it

Since logarithms

follows that

log(2x 3)=

log 2

2

+ log 3.

= 10

2x3 = 10°-

-

3010

3

,

= 10°- ml = 10 -" = -

x 10 "1

8010

-

4

81

The logarithm ofaproduct is equal to

6.

the sum, of the logarithms

of the factors. II. log

|

= log 3 - log 2. 2 = 10°- 4m

3

-=-

-=-

10

-

3010

= 10

-

1 ' 61

= 1.5.

The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. III.

log2 3

= 31og2. 2 = no 3

0,3010 8 )

= 10

-

9080

=

8

The logarithm of a power of a number is equal to the logarithm of the number multiplied by the exponent of the power. IV. log

VI = log 3* = i log 3. V3 = 3* = (10 " )* = 10 -

4

1

-

2386

= 1.732.

The logarithm of the root of a number is equal to of the number divided by the index of the root. PROBLEMS 1.

Multiply 28.34 by 3.376.

= = log product = product = log 28.34

1.

log 3.376

0.

the logarithm

COMMON LOGARITHMS

125

Before finding the mantissas from the table always make out an This saves time and prevents mistakes. Keep the signs of equality and the figures exactly in columns.

outline as above.

Solution.

= = product = product =

log 28.34

log 3.376

log

0.5284 1.9808 95.68.

we have 28^ x3J =

As

a rough check

2.

Multiply 1.251 by .6453.

Solution.

1.4524

94.

= 0.0973 = 9.8098 - 10 log product = 9.9071 - 10 product = .8074 .65 X 1\ = .81. log 1.251 log .6453

Rough

check.

Divide 31.87 by 641.2.

3.

Solution.

= 11.5034 = 2.8070 log quotient = 8.6964 quotient = .04970. 32 640 = .05. log 31.87

10

log 641.2

Rough

check.

10

-f-

Since the characteristic of the logarithm of the divisor

is

larger than the characteristic of the logarithm of the dividend,

10

is

dend.

added to and subtracted from the logarithm of the diviNote that the quotient has four significant figures

(see sect. 2).

The

zero

must be written

at the right to

that the division has been carried out to four figures. 4.

Divide .8354 by .04362.

Solution.

= 9.9219 - 10 = 8.6397-10 log quotient = 1.2822 quotient = 19.15. .84 .044 = 19. log .8354

log .04362

Rough

check.

-f-

show

126 5.

APPLIED MATHEMATICS Find .6874s

COMMON LOGARITHMS The base

28.

8.482

Find

ft.

of a triangle

72.14

is

ft.

127

and

its

altitude is

its area.

29.

Find the area of a square whose side

30.

The parallel sides of a trapezoid are 69.14 ft. and 38.15

If the altitude

is

12.83

ft.,

71.18 yd.

is

ft.

find the area.

Find the surface and volume of brass cylinders and wooden blocks, and so on.

31.

prisms,

32. .Find the area of the blackboard in square meters. 33.

Find the area of the

34.

Find the area of the ground covered by the school

athletic field.

buildings.

Find the area of the block in which the school building

35.

stands. 36. Construct the logarithmic curve.

The area

37. is

58.12 yd.

of a rectangle

Find

its

is

1689

and the length

sq. yd.

width.

38.

Find the side of a square whose area

39.

The volume

of a cube

is

2861

is

cu. in.

77.83 sq.

ft.

Find the length

of an edge.

What

40.

172.8 sq.

is

the diameter of a piston which has an area of

in. ?

41. Find the diameter of a circular plate of iron of the same weight and thickness as a rectangular plate 3 ft. 4 in. by 2 ft. 8 in. 42.

Find

.

A

its

steel shaft is 3.5 in. in

weight

if

1 cu.

in.

diameter and 12

of steel weighs .283

ft.

lb.

9

in. long.

CHAPTER XI THE SLIDE RULE Use of the slide rule.* In ordinary practical work it is make measurements and carry results in computations only to three or four significant figures. With the slide rule multiplications and divisions can be performed mechanically to the degree of accuracy required in this work. The slide rule is 75.

usual to

A

;

THE SLIDE RULE the 301st graduation

;

log 3

=

129

.477, therefore

the 477th graduation; and so on for

all

3 was placed at

the integers from 1

to 1000.

To read

the numbers from 1 to 1000 we must go over the from left to right three times. Thus we read first 1, 2, 3, 10 then beginning at 1 again and calling it 10, we read 10,

rule •

,

;

20, 30,

we read

100

,

then beginning at 1 again and calling

;

100, 200, 300,

77. Operations

•

,

it

100,

1000.

with the

slide rule.

It is not difficult to

learn to use the slide rule if at first the operations are per-

formed with small numbers. Whenever in doubt about any operation perform it first with small numbers. I.

set 1

Multiplication.

C on

that this

3D

is

;

Multiply 3 by

then under 2

C

2.

Move

the slide so as to

read the product 6 on D. Note

simply adding logarithms.

To find the product of two numbers, set 1 C on one of the numbers on D, and under the other number on C read the product on D. Sometimes in multiplying we must use the 1 at the right end of scale C. Thus multiply 84 by 2. Set 1 at the right end of scale C on 84 D, under 2 C read 168 on D. We use the 1 at the left end or the right end of scale C according as it brings the second factor Over scale D. In the example above, if we had set 1 at the left end of scale C on 84, then 2 C would have been

off scale D.

The decimal point

is placed by inspection. Thus, multiply by 1.8. Set 1 C on 18 D, under 125 C read 225 on D. But making an approximate multiplication mentally, 12 x 2 = 24 hence we know that there are two integral figures in the prod-

12.5

uct, giving 22.5 as the result.

In

all

rule the decimal point can be placed

operations with the slide

by making an approximate

mental computation. II.

by 2. Set 2 C on 8 D, under 1 C read Note that this is simply subtracting

Division. Divide 8

the quotient 4 on D. logarithms.

APPLIED MATHEMATICS

130

set the divisor on scale C C read the quotient on scale D.

To divide one number by another, on the dividend on scale D, under 1

The decimal

point

3.44 by 16. Set 16

215 on tient

D

we

see that 3

-s-

16

=

1

C

about

read the quotient .2

;

hence the quo-

is .215.

Combined multiplication and

III. 24'

of

but

;

Thus divide

placed by inspection.

is

C on 344 D, under

v 9 Set 6

„

C on 24

division.

D, under 9

C

Find the value

read the result 36 on

o

Study this operation till the separate parts are seen clearly and understood. First the division of 24 by 6 is made by setting 6 C on 24 D, under 1 C we might read the quotient but we want to multiply this quotient by 9. As 1 C is already on this quotient we have only to read the product 36 on scale D.

;

D under An

9 C.

important problem under this case

is

to find the fourth

term of a proportion. T;hus, in the proportion 6 24 :

24

*

=9

:

as,

x = -g— 9

Hence to find the fourth term of a proportion, set the term on the second, under the third read the fourth.

first

IV. Continued multiplication and division. Here for convenience

we need the runner. This

piece of glass

which has a

on

line

is

a sliding frame carrying a

it

perpendicular to the length

of the rule. 1.

Find the value of 3 x 8 x

5.

on 3 D, set runner on 8 C, set 1 C at the right on the runner, under 5 C read 12 on D. Hence Set 1

C

at the right

3x8x5 = 120. 54 2.

Find the value of

-z

3x6

•

;

THE SLIDE RULE Set 3

C

1

C on 54

131

C on runner, under Note that we have simply made two

D, set runner on 1 C, set 6

read result 3 on D.

divisions.

Find the value of

3.

Set 24

C

1

—24

—

•

x 6

C on 15 D, set runner on 48

C, set 6

C on runner, under

read result 5 on D.

4.

—

Find the value of

C on 8 D, set runner on 1 C, set 1 C at right end of on runner, set runner on 9 C, set 1 C on runner, under 4 C read result 9 on D. In a similar manner any number of continued multiplications and divisions may be performed. V. Squares and square root. The graduations on scale A at Set 32

slide

'

the top of the slide are arranged so that the square of every

number on

C

it on scale A. Thus and above 25 is 625. On scale A the distances of the numbers from 1 at the left end of the scale are proportional to the logarithms of the numbers as on scale C but it is easier to learn to use scale A by noticing its relation to scale C. We read from left to right 1, 2, 3, •, 10, 20, 30, then beginning at 1 again and calling it 100, we read •, 100 100, 200, 300,--., 1000, 2000, 3000,- •, 10,000. The first 4

above 2

scale

is 4,

stands directly above

above 3

is 9,

•

•

is

;

either 4 or 400, that

second 4

is

is,

either the square of 2 or 20

either 40 or 4000, that

is,

;

the

either the square of 6.32

or of 63.2.

To square any number, find the number on its

square directly above

To

it

scale

C and

read

on scale A.

its

any number, find the number square root directly below it on scale C.

scale is

very convenient when multiplying or

extract the square root of

on scale

A and read

The upper

dividing by square roots, finding the area of circles, and so on.

APPLIED MATHEMATICS

132 1.

Find the value of 8 V3.

Set 1 13.8E

C

at right

end of

scale

on 3i, under 8 C read

result

THE SLIDE RULE 7.

To make 865

lb.

of iralty metal, used for parts of

engines on naval vessels, 752.5

and 69.2

133

lb.

of copper, 43.3

of tin were melted together.

lb.

lb.

of zinc,

Find the per cent

of each metal used. 8.

17

lb.

of copper, 85

lb.

of tin, 595

of antimony were melted together to metal. 9.

What

lb.

of lead,

make 850

10.

If sea water contains 2.71 per cent of salt,

The

lb.

of type

per cent of each metal was used ?

tons of sea water must be taken to give 100

oak beams

and 153 lb.

lb.

how many

of salt ?

safe load in tons, uniformly distributed, for whiteis

given by the formula

W

= the safe load in tons, b = the breadth in inches, where d = the depth in inches, and I = the distance between the s in inches.

Construct a curve to show the safe load in tons for whiteoak beams having a breadth of 3 in., distance between s

13

ft.,

11.

and depth from 3 If

w = the

in. to

15

in.

weight of 1

lb.

of

pended in water, and

any substance when

s its specific gravity,

1

1

—w

or

w=

sus-

then

s-1

Construct a curve showing the weight of substances susspecific gravity varying from .5 to 15.

pended in water, the

CHAPTER

XII

ANGLE FUNCTIONS 78. Angles.

Suppose

AP

AP and AM be

Let two lines

to revolve about the point

the amount of turn, indicated by the arrow,

The amount of turn

expressed in degrees.

is

gives an angle of 360°, a half turn 180°,

coincident.

A away is

from

called

A

an

AM; angle.

complete turn

and a quarter turn

In this chapter we will not consider angles greater than

90°.

90°.

Fig. 55

The is

line

AM which. marks

the beginning of the revolution

called the initial line; the line

of the revolution

is

AP

which marks the ending

called the terminal line of the angle.

from any point B in the BC is dropped to the the right triangle formed

79. Triangle of reference.

If

terminal line of the angle a perpendicular initial line, is

called the triangle of reference for

the angle.

The perpendicular

called the opposite side

;

A C,

BC

is

the part

of the initial line cut off by the perpendicular,

side; and

minal line

is

called

the

AB, that part which belongs

adjacent

Fig. 56

of the terto the triangle of reference,

the hypotenuse. 134

is

called

c

ANGLE FUNCTIONS

135

80. Sine, cosine, and tangent of an angle. Given the angle A. Construct the triangle of reference, and represent the lengths of the sides by a, b, and c, set opposite the angles A, B, and C respectively.

AC = AB

BC _a

AB~ _ opposite =

side

hypotenuse A (by def-

sin

is

usually

approxi-

c

_

adjacent side

=

cos

inition).

This ratio, called the sine of angle A, is a pure number which

BC = AC

b

hypotenuse A (by def-

a b

_

opposite side

=

adjacent side tan A (by def-

inition).

This

ratio, called

inition).

the

cosine of angle A,

a pure is

is

number which

usually

approxi-

This ratio, called the tangent of angle A, is a pure number which is usually approximate

mate and expressed

mate and expressed

and expressed as

as a decimal.

as a decimal.

decimal.

These the angle

ratios sin A, cos

A

A, and tan

A

a

are called functions of

because they change in value as the angle changes.

There are other functions of an angle, but as these three seem more important the discussion will be limited to them.

to be the

EXERCISES 1.

Make an

angle

A and

construct the triangle of reference.

Letter as before, and measure the sides as possible in millimeters.

Use the

to find the values of sin A, cos A,

a, b,

and

c as

results of the

accurately

measurement

and tan A. Carry the

sions as far as the errors in the approximation justify,

no

divi-

and

farther.

Make another

angle A' which differs from A. Calculate and tangent in the same manner. Compare the values of the two sines, the two cosines, and the two tangents. If you were to continue the experiment, you would find that 2.

its sine,

cosine,

the ratios change in value every time the angle changes in

size.

.

APPLIED MATHEMATICS

136 3.

Make an

angle and drop perpendiculars from various

points on the terminal line to the initial line.

Any

one of

the-

may

be considered a triangle of reference for the angle. Find sin A from each triangle of reference. Compare the values. Should they all be equal ? Why ? Similarly right triangles

for cos 4.

A and

In

a

AB = 6 in., be chosen at

tan

A

triangle

of

reference

and A C = 5 in. ? Why random ? Why ? Could

one side be chosen at random?

Why? 81. Functions of 45°.

an angle of of reference

1 unit long. 1 unit long ?

V2 units

45°,

and

Construct

in the triangle

either iC or BC Why is the other side Why is the hypotenuse

make

long ?

ABC ?

could

BC = 2 in.,

Could any two

sides

ANGLE FUNCTIONS 83. Functions of 60°.

the triangle of reference 1 unit long. is

Why

BC V3 units sin 60°

is

Construct an angle of 60°, and in

make the

the hypotenuse

side

AB

„. ano _., cos 60° = \

to 60°,

2 units long ?

= =

This ratio

=

i (1-732) .866.

*__„,«, 60° tan

Why

_V5

.500. is

exact.

the functions of 60° can be

found from the triangle of reference for 84.

A C adjacent

long ?

= V3

Show how

137

30°.

Table of angle functions. The funchave been calculated and

tions of angles

In solving problems the funcfrom the table. The functions of a few angles, 30°, 45°, 60°, 90°, should be memorized. tabulated.

tions of the angle are taken

PROBLEMS 1.

A

man

standing 110

ft.

from a tree on level ground

finds the angle of elevation of the top of the tree to be 37° 20'.

How high is the tree, and how far is the man from the top of it ? Solution. Given

APPLIED MATHEMATICS

138 2.

A

3.

A

track has

railroad

How many

horizontal.

ladder 24

the ladder

4

is

ft.

ft.

4

in.

a uniform slope of 5° to the

feet does a train rise in going a mile ?

long rests against a wall. The foot of from the wall. Find the height of the

top of the ladder. 4.

The shadow

of a tree

elevation of the sun 5.

A

ship

is 42°.

is

38

ft.

long

when

sailing northeast 12 mi. per hour.

is

the angle of

Find the height of the

tree.

How fast is

she sailing east ? 6.

A

stick 8

ft.

long stands vertically in a horizontal plane,

and the length of the shadow elevation of the sun ? 7.

What

is

The top

ft.

9.

of a lighthouse

the angle of

if it rises

118

ft.

152

is

is

ft.

12°

above sea

15',

how

far

level.

If

from the

is it ?

The chord

is 4.4 in. and it subtends Find the radius of the circle.

of a circle

center an angle of 38°. 10.

is

along the path ?

the angle of depression of a buoy lighthouse

What

ft.

the slope of a mountain path

in a distance of 835 8.

6

is

At a point 212

at the

from the foot of a column the angle column is found to be 24° 28'. the height of the column ? ft.

of elevation of the top of the

What 11.

his

is

A

man

shadow

is

6

ft. tall

17

ft.

stands 4

long,

what

is

ft.

6

in.

from a lamp-post.

If

the height of the lamp-post

?

12. A cable is attached to a smokestack 10 ft. below the top, and to a pile 42 ft. from the foot of the stack. If the cable makes an angle of 62° 20' with the horizontal, find the height

of the stack.

From

vessels appear in line.

ft. above sea level two If their angles of depression are 4° 20'

and 2°

how many

13.

the top of a lighthouse 160

45' respectively,

miles are they apart ?

ANGLE FUNCTIONS 14.

As the angle

35° 15' to 64° 25', steeple 15.

120

ft.

139

of elevation of the sun increases

how many

feet does the

shadow

high decrease ?

r

In the gable shown in the figure

angle

BAF=60°,

BG =

6

and FE, and AC. ft.,

GFE =

angle

GE =

4

ft.

from

of a church

/v^— -ll^S/> ~"

30°,

//

Find AF,

\

j

\^°

—

~

'

j

Fig. 62

16.

The base

AC

of an isosceles

and the equal sides AD and CB make angles The altitude is 40 ft. Compute the length of the upper base and the area. Draw to scale and check. trapezoid

is

100

ft.,

of 60° with the base.

17.

The

pitch of a roof (angle which the rafters

the horizontal)

is

32°.

If the house

is

22

ft.

make with

wide, find the

length of the rafters and the height of the gable. 18.

of

A

building 80

ft.

long and 40

ft.

wide has each side Find the area of

roof inclined 40° to the horizontal.

its

the roof.

Two towns A and B are at opposite ends of a lake. It is that a station P is 3 mi. from A and 2 mi. from B. If angle PAB = 3A°30' and angle PBA = 62° 40', find the

19.

known the

distance between the towns. 20.

solve

Make

a height or distance problem of your

own and

it.

85. Logarithmic solutions. In the preceding problems the numbers involved consist of only two or three figures hence there would be little or no time saved in using logarithms. However, "when there are several figures in the numbers, and there are three or more multiplications or divisions, logarithms ;

should be used.

The logarithms of the angle functions are found in exactly way as are the logarithms of numbers. Thus, find

the same

log sin 18° 26'.

APPLIED MATHEMATICS

140

Mantissa log sin 18° 30'

—

mantissa

= 5015 - 4977 = 38. 38 x .6 = 23. 4977 + 23 = 5000. log sin 18° 26' = 9.5000 - 10.

log sin 18° 20'

.-.

The

sine

and tangent of an angle increase

increases, hence the difference for the minutes is

mantissa of the smaller angle taken from the It is to be noted that the cosine of

angle increases

;

as

the angle

added

to the

table.

an angle decreases as the

hence the difference for the minutes

is

to be

subtracted instead of added.

Thus

find log cos 24° 48'.

— mantissa = 9584 - 9579 = 5. 5 x .8 = 4. 9584 - 4 = 9580. log cos 24° 48' = 9.9580 - 10. log tan x = 9.5946 — 10, find x.

Mantissa log cos 24° 40'

log cos 24° 50'

.-.

Given

21° 20' 21° 2 21° 30'

5917 5946 5954 37)290(8

.-.

x

= 21° 28'.

This work should be done mentally. In class find logarithms of the functions of

arithms

many

angles,

and the angles from the

log-

of the functions, as quickly as possible until this can

be done readily.

The

Why ? 8 — 10,

and cosine of an angle are always less than 1. Hence the characteristic of the logarithm is 9 — 10, and so on. The — 10 is not printed in the table, but

sine

should be written in computation.

.

ANGLE FUNCTIONS

141

PROBLEMS In the right triangle ABC,

1.

C= A=

given

c

Find a and

=

90°,

28°

34',

48.32

ft.

b.

Solution.

-

=

-

sin.

=

cos A.

c

= log c = sin 4 = log a = a = a

log

c sin yl

b

log cos

A 6

as above.

a

A

c

Check.

It

out an outline

"a

may

but an answer

be as is

= ccosA. = 1.6841 cos A = 9.9436 log b = 1.6277 b = 42.43 ft. b

log c

10

log

much work

10

problem as to solve it is known to be knowing how to work problems to check a

absolutely worthless unless

What is the advantage of you cannot get correct results ?

correct. if

make

c

= csin^4. log c = 1.6841 sin A = 9.6796 loga = 1.3637 a = 23.11 ft. a

it,

cos A.

log 6

Before looking up any logarithms always

log

c

log c

= c 2 (Pythag. th.). = c'*- V = (c-b)(c + b). c-b= 5.89 log = 0.7701 c + b = 90.75 log = 1.9579 log a 2 = 2.7280 log a = 1.3640 a = 23.12. a2

+

i

2

a'*

A

difference of 1 in the last figure

logarithms are only approximate.

may

be expected since the

:

APPLIED MATHEMATICS

142 2.

Two

trees

M and N are on opposite

NPM is 3.

sides of a river.

MN

NP at

line

A

right angles to is 432.7 ft. long and the angle 52° 27'. What is the distance from to ?

From

M N

the top of a building 156.4 is 18° 46'.

depression of a street corner

ft.

high the angle of

Find the horizontal

distance from the street corner to the building.

To

Auditorium tower a distance was measured from the foot of the tower. and the angle of elevation of the tower was found to be 40° 57'- Find 4.

of 311.2

find the height of the

ft.

the height of the tower.

Solve the following right triangles, two parts being given 5. 6. 7.

8.

13.

= 146.8, b = 203.3. = 49.74, A = 53° 38'. c = 94.53, B = 62° 51'. e = 436.5, A = 74° 11'. a

9.

b

Find the

side of

= 110.9, a = 64.21. = 8.226, c = 12.15. c = .02936, a = .01153. a = .9681, A = 42° 17'. e

10. b 11. 12.

an equilateral triangle inscribed in a

circle of radius 52.18 in. 14. is

The

14.26

side of an equilateral triangle inscribed in a circle

in.


circle.

15. If a side of a regular pentagon

is

30.24

in.,

find the

radius of the circumscribed circle, and the apothem. 16.

A

11.32

in.

17.

What 18.

What 19.

regular pentagon

The apothem is

inscribed in a circle of radius

of a regular polygon of 12 sides

is

21.26

ft.

168.4

ft.

the perimeter ?

The perimeter is

is

Find a side and the apothem of the pentagon.

of a regular octagonal tower

is

the area of the base of the tower ?

A regular octagonal column is cut from a circular cylinder

whose diameter

is

18.32

in.

Find the area of a

cross section of

the column. 20.

A

28.43

ft.

same

circle.

side of a regular

Find a side

hexagon inscribed

in a circle

is

of a regular decagon inscribed in the

:

ANGLE FUNCTIONS Area

86.

ABC, h

C

143

In triangle

of triangles.

the perpendicular from

is

toe.

Area triangle

ABC = \ base x

altitude

.

= I ch = \cb sin .4. The area, of a triangle equals one half the product of two sides and the sine of the included angle.

PROBLEMS 1.

c

=

Find the area of a triangle ABC, given a 76.31 ft., and B = 29° 18'.

Solution.

= 42.84

ft.,

= ac sin B. = 1.6318 log c = 1.8826 log sin B = 9.6896 - 10 log 2 area = 3.2040 2 area = 1600. area = 800 sq. ft. 2 area

log a

Find the area of the following

triangles.

Check by finding

the area twice, using different angles 2. 3.

4. 5.

6. Find the area of a y = 381.3ft., and Z = 51° 7.

The

c

What

is

ft.,

a;

24'.

vertical angle of

the equal sides are 16.46 8.

= 98.32, A = 17° 43', C = 60° 36'. c = 54.38, B = 40° 10', C = 36° 47'. c = 1,610, A = 53° 17', B = 62° 24'. e = 1.435, B = 42° 55', C = 24° 22'. triangle XYZ, given = 184.2

= 34.36, b = 110.5, a = 88.48, b = 58.59, a = 1.432, b = 1.583, a = 3.207, b = 2.367, a

ft.

an

isosceles triangle is 75° 18'

long.

Find the area of the

the area of a parallelogram

if

two adjacent

are 243.6 yd. and 315.4 yd. and the included angle

is

and

triangle.

sides

35° 40' ?

APPLIED MATHEMATICS

144

Two

make an angle of 53° 18' with each other. between them has a frontage of 286 ft. on one street and 324 ft. on the other. Draw to scale and find the area 9.

streets

The corner of the

lot

lot.

Two

an angle of 21° 25'. From a point from the crossing how must a fence on be run so that the inclosure shall contain 10 A. ? 10.

one of

11.

railroads cross at

them 100

The survey

rd.

of a field gave the following data:

EA = 420 ft. EB = 865 ft. EC = 875 ft. £2)

=650

"

ft.

AAEB = 42°. Z BEC = 36°. Z CED = 20°. Draw

the field to scale and

Fig. 65

find its area. 12.

A

surveyor set his transit over the corner A of a field the angle DAC = 40° 12', and angle CAB =

A BCD and found 70°

AD

54'.

to scale

87.

is

52.8

AC is

86.3 rd., and

AB is 38.4 rd. Draw

field.

of sines. In the triangle ABC from the vertex C to the side c.

Law

dicular

h —

= sm A .

,

b

By

rd.,

and compute the area of the

h i and a

let

h be the perpen-

= sin BD •

division, *

a

_ sin A

b

sin

a sin

B

(1)

b

A

sin B,

What algebraic operations were used to derive (2) from (1) ? What theorem in geometry could be used for this purpose ?

ANGLE FUNCTIONS By

dropping a perpendicular from

A

145 a we

to

may

obtain, in

a similar manner, sin

a sin

Law

C

sin

b

A

sin

B

e

B

sin

C

of Sines. In any triangle the sides are proportional

to

the sines of the opposite angles.

When

a side and two angles of a triangle are given

find the other

two sides by

we may

this law.

PROBLEMS 1.

In a triangle

36.74.

Find

b

a

b

B

sin

given

A =

36° 56',

180°

-

(A + B)

=

72°

70° 58'. c

A

B=

c.

C=

Solution. sin

ABC

and

a

6',

and a

=

.

APPLIED MATHEMATICS

146

7. On board two ships half a mile apart it is found that the angles subtended by the other ship and a fort are 84° 16' and 78° 38' respectively. Find the distance of each ship from the

fort. 8.

P

M and N are stations

on two hilltops 3684

ft.

apart, and

NMP

MNP

and a station on a third hill. The angles are observed to be 50° 42' and 63° 24' respectively. Find the disis

tances

is

and NP.

Law of cosines.

88.

h

MP

C

the perpendicular from

In triangle on

In triangle on

to

c.

= h + b' - = cos^l, b' = b cos A.

left, b

and

a

In triangle ABC, 2

2

2

,

(1)

(2)

right,

a2 Substituting (2),

=h =h =h =b

2 2

+ (c - b') + c -2b'c + V + + c - 2 be cos A. + c - 2 be cos A 2

2

2

.

2

b'

2

2

2 2 a2 Similarly, by dropping perpendiculars from

Substituting (1),

b c

Law equal

to

2

2

sum

2

2

of Cosines. In any the

A and B we

get

- 2 ac cos B.

=a +a = a + b -2ab cos C. 2

2

triangle the square of

any

of the squares of the other two sides

the product of these two sides

and the

side

is

less twice

cosine of the included

angle.

PROBLEMS 1.

Find a

in the triangle

ABC, given b

=

6

in., c

A = 29° 15'. Solution.

= J 2 + c 2 -2bccosA = 36 + 25-2x6x5x = 8.65. a = 2.9 in.

a2

.8725

5

in.,

and

:

ANGLE FUNCTIONS In triangle ABC, find A. if a

2.

a2

=

COS.4

=

Solution.

b2 &2

+ +

147

= 7, b = 8, c = 9.

— 2 be cos A. C 2 _ a2 c2

2 6c

_

64

+

81

-49

2x8x9 = .6667. A = 48° 11'. Find

3.

B and C

Problem 2 and check

in the triangle in

by-

adding the three angles. Solve the following triangles and cheek by drawing to scale or otherwise 4.

a

5.

a

= 10, b = 12, c = 14. = 4, 6 = 5, c = 6.

6.

b

7.

a

= 21, c = 19, A = 48°57' = 14, 6 = 12, c = 60°.

8. Two ships leave a dock at the same time. One sails east 12 mi. per hour and the other northeast 14 mi. per hour. How far will they be apart at the end of 5 hr. ? 9. From a point 5 mi. from one end of a lake and 4 mi. from the other end, the lake subtends an angle of 56° 8'. What is

the length of the lake ?

A and B are two stations on opposite sides of a mounand C is a station on top of the mountain from which A and B are visible. If CA = 4.2 mi. and CB = 3.1 mi., and angle 10.

tain,

ACB = 88° 12', find the

distance from

A

to B, the three stations

being in the same vertical plane. 89. Triangle of forces.

boom

is

acting

The weight

held in position by three forces

downward

(b)

;

:

W

at the

the tension (pull) in the

thrust (push) of the boom.

end of the

(a) the force of gravity

The tension

tie

;

The

triangle is proportional to the lengths of the sides.

sion in the mast

tension per foot Fig. 68, if

is

is

always taken equal to the load

in the mast

W

;

the same in each side of the triangle.

W = 2000

(c)

the

in each side of the ten-

and the Thus in

lb., AB = 10 ft., and BC = 16 ft., the tension AB = 2000 lb. and the tension per foot = 200 lb.

APPLIED MATHEMATICS

148

= 16 x 200 = 3200 lb. AC = VlO + 16 = V356 = 18.9 ft., and the tension AC = 18.9 x 200 = 3780 lb.

Therefore the co mpre ssion in the boom

The in

2

tie

3

Check.

20002 32002

2

= 4,000,000 = 10,240,000 14,240,000

3780

2

= 14,290,000

A

Fig. Exercise.

Put two screw eyes

Simple Crane

in the wall 80 cm. apart

and

construct a model of a crane, using a meter stick, string, and a

spring balance,

as.

shown

in Fig. 69.

Compute the

tension for

eocm aoon.

Fig. 69

and check by the readings of the spring After a weight has been attached the string should be shortened enough to make the string or the meter stick perpendicular to the wall in order to form a right triangle. different weights

balance.

PROBLEMS 1.

boom

The mast is

of a crane

is

12

ft.

long and the

tie

horizontal and s a load of 2400

tensions in the

boom and

tie.

The ft. Find the

18

lb.

ANGLE FUNCTIONS The tie of a crane is horizontal. boom is 30 ft. long, find the tension

If

2.

the

tie for

149 it is

24

ft.

long and

in the mast, boom,

and

a load of 4 T.

3. The tie of a crane makes an angle of 30° with the mast, and the boom is horizontal. If the boom is 20 ft. long and the load is 3000 lb., find the tension in the mast, tie, and boom. 4.

The boom

of a crane

in the mast, boom, 5.

The boom

zontal the tie

4200

lb.,

be lifted 6.

is

is

The

of 40° with the mast.

and

tie for

of a crane

30

ft.

16

ft.

long and makes an angle

tie is horizontal.

is

long.

Find the tension

a load of 2 T.

20

ft.

long,

and when

it is

hori-

If the tie can stand a strain of

find the greatest load that can

when the boom

The bracket

BCD

carries a load of

Find the

4001b. at D.

horizontal.

is

stresses in

BC,

CD, and BD. arc lamp weighing 20 lb. is 7. An hung on a pole, as shown in Fig. 71. Find the stresses in MP and NP. 8.

A

weight of 96

cord which

A and

lb.

Fig. 70

M

attached to a

is

secured to the wall at a point

is

pushed out from the wall by a horizontal stick BC. If ^4C = 6ft. and angle

is

BA C =

38°, find the tension in

AB

/v

Fig. 71

and the pressure on BC. 9. A canal boat is kept 20 ft. from the towpath and the towline is 72 ft. long. If there is a pull of 144 lb. on the line, what

Fig. 72 is

the effective pull

?

Solution. Let C, Fig.

72,

be the position of the canal boat.

AB = V72 - 20 = 2

•l^4.:

69.2 x 2

= =

2

lb.,

138.4

2

69.2

ft.

the tension per foot in lb.,

the effective pull.

AC.

APPLIED MATHEMATICS

150

10. The pull on the towline of a canal boat is 4001b. and the line makes an angle of 10° with the direction of the boat.

How much

of the pull

How much

effective ?

is

is

at right

angles to the direction of the boat ? 11. A boat is pulled up the middle of a stream 60 ft. wide by two men on opposite sides, each pulling with a force of 100 lb. If each rope, attached to the bow of the boat, is 40 ft. long, find the effective pull on the boat.

12.

Each

of two horses attached to a load

force of 200

lb.

each other, what

is

pulling with a

If they are pulling at an angle of 60° with is

the effective pull on the load ?

Attach two spring balances to the wall, as shown in Kg. 73, with 10 or 12 ft. of cord between them. At the center of the cord attach an 8-lb. weight. Read each bal13.

ance for the tension in

Suppose

AC

^C = 6ft.

Compute the

stress

Solution.

and in AC.

and BC.

DC =

4ft.

= 4 lb., stress in DC. = 1 lb. per foot, stress 1x6 = 6 lb., stress in A C.

£

of 8

\

in

DC.

Compare with result of the experiment. Make other experiments with different lengths of cord until the reason for the s' a method of computation is understood, a < s' C

A man

weighing 180 lb. sits in hammock 12 ft. long. If the s are 10 ft. apart, find the pull on the hammock. 14.

the center of a

Solution.

CD \ of 180 90

-.

V6 - 5 = 2

2

E3 Fig. 74

3.32

ft.

=

90 lb., pull in CD.

=

pull per foot.

=

163

3.32

90 x 6 3.32

163

lb.

= pull

lb.,

pull in y

A D.

on hammock.

ANGLE FUNCTIONS cos x

Check.

x Pull in

15.

Two

=f = .8333. = 33° 34'.

CD =

Fullin

AD SmX

Pull in

CD = sin

-

a;

x pull in

AD

= .553 x 163 = 90.2 lb. 2 = 180.4 lb., weight

90.2 x

151

of the

man.

horses attached to a load are pulling with the same

force at an angle of 60° with each other. effective pull

on the load

400

is

lb.,

If the combined

how many pounds

is

each

horse pulling ?

Connect two light strips of wood 60 cm. long, AB and BC by a hinge at B, and put casters at A and C. Put a cord and spring balance between A 16.

(Fig. 75),

and

C, as

shown

in the figure.

the frame vertical, measure

AC, and read the

Hold

BD and

balance,

when

BD =

48 cm. and AD = 36 cm. Attach an 8-lb. weight at B and make AC = 72 cm. Read the balance, and

Fig. 75

subtract the first reading to get the tension in 8-lb.

weight.

=

in

AC

AC

as follows

due to the :

y ^ lb., tension per centimeter in BD. 3 lb., tension in AD. TV X 36 tension in A C 3 lb.

£g

.'.

Compute the tension 1

= =

Compare with the

result of the experiment.

Make

other ex-

periments with different weights and distances AC, until the reason for the method of computation 17.

A pair of rafters

at the ridge.

the building

The is

30

is

understood.

s a weight equivalent to 800 lb. is 30° and the width of

pitch of the roof ft.

foot of each rafter.

Find the tension in the

tie

through the

APPLIED MATHEMATICS

152 18.

long.

The width

of a house

is

24

ft.

and the

rafters are 16

If the rafters a weight equal to

600

lb. at

ridge, find the stress in the rafters. 19.

A

bridge truss

a weight of 300 tally.

CD =

lb.

The span 10

ft.,

ABC

s

per foot horizon-

30

long.

If

find the stresses in

AC

is

and AB. (The load

at

ft.

D

equals one

half the total load.) 20.

ABC

(Kg. 77)

is

an inverted

AB = 20 ft., and the angles CAB a,nd ABC = 40°. If the load at D is 4 T., find the stresses in king-post truss.

A C and AB.

Fig. 76

ft.

the

CHAPTER

XIII

GEOMETRICAL EXERCISES FOR ADVANCED ALGEBRA 90.

A

figure should be

drawn

for each exercise, letters or

numbers put on the lines in the figure, and the equations set up from the figure. Check by drawing to scale and measuring the required parts.

The

first

exercises involve square roots,

since radicals are reviewed early.

Some

of the exercises should

be worked out in notebooks, with emphasis placed on accuracy in

drawing and neatness in arrangement. 1.

13.

to Construct a graph for the square's of numbers from Units horizontal, 1 large square = 1 vertical, 1 large :

square = 10.

;

What is the equation of the V&25, VlO, Vl2, and Vl2i5

V6, V7J5, and check by the graph. 2. Find the diagonal of a square whose

curve ?

Find v2,

to three decimal

places

side

is

12

(s).

Eig. 78

Solution.

= 12 2 + 12 2 (Pythag. th.) = 288 d = V288 = Vl44 x 2 = 12V2 = 16:97.

rZ

2

153

=

s2

+

s2

=2s 2 d

= sV2.

(Pythag. th.)

APPLIED MATHEMATICS

154

Find the side of a square whose diagonal

3.

The side of an altitude and area. 4.

The altitude and area.

5.

side

of

an equilateral triangle

The area of an and altitude.

6.

side

5

is

(d).

Find the

equilateral triangle is 4(s).


6(A).

Find the

24(«).

Find the

is

is

7.

Find the area of a regular hexagon whose side

8.

Find the area of a regular hexagon whose apothem

is

3(s). is

2(h). 9.

area

is

10.

Find the side and apothem of a regular hexagon whose 36 (a).

A star-shaped figure is formed by constructing equilateral

triangles outwardly on the four sides of a square.

of the entire figure 11.

is

If the area

100, find a side of the square.

Squares are constructed outwardly on the sides of a

regular hexagon.

If the area of the entire figure

is

72, find a

side of the hexagon. 12.

From a

square whose side

formed by cutting

off

13.

The edge

14.

The diagonal

is

the corners.

of a cube

Find a diagonal.

is 5(ie).

of a cube

is

8

12 (s) a regular octagon is Find a side of the octagon.

(d).

Find an edge.

Find the diagonal of a rectangular parallelepiped whose edges are 4, 5, and 6 (a, b, and c). 15.

16.

Find the side of an equilateral triangle whose area equals is 6 V60.

the area of a square whose diagonal 1 7. is

£ ab 18.

is

sides of a triangle are a

when If

the included angle

\

If

is

and

two

the included angle

is

sides of a triangle are a

V2 ab when

Show

b.

that the area

30° or 150°.

two sides of a triangle are a and

^ V3 ab when 19.

is

Two

the included angle

is

b,

show that the

area

60° or 120°.

and

b,

show that

45° or 135°.

the area

GEOMETRICAL EXERCISES

155

20. The sides of a triangle are 30, 60, and SO (a, b, and c). Find the segments of each side formed by the bisector of the

opposite angle.

The shadow

cast upon level ground by a certain church 27 (37) yd. long, and at the same time the shadow of a vertical rod 5 (7) ft. high is 3 (6) ft. long. Find the height 21.

steeple

is

of the steeple. 22.

apart.

The footpaths on the opposite sides of a street are 30 ft. On one of them a bicycle rider is moving at the rate

of 15 mi. per hour.

If a

man on

the other side, walking in the

opposite direction, regulates his pace so that a tree 5

path continually hides hiin from the does he walk ? his

rider, at

ft.

what

from rate

23. One side of a triangle is divided into two equal parts and through the point of division a line is drawn parallel to the base. Into what fractional parts is the triangle divided ? Similarly, when the side is divided into 3, 4, 5, n equal parts ? •

24.

•

Find the side of an equilateral triangle is 2 (x) in. from the vertex.

•

,

if

the center of

gravity

25. What part of a triangle lies between the base and a line through the center of gravity parallel to the base ?

26.

One

side of a triangle

is

10 (s)

in.

Where must

a point

be taken in the given side in order that a line drawn through it, parallel to another side, will divide the triangle into two areas 27.

whose

ratio is 3

The bases

altitude

is

:

4 (m n) :

?

of a trapezoid are 16 and 10 (b r and 5 2) and the

6(A).

Find the area of the triangle formed by

producing the nonparallel sides of the trapezoid. 28. Find the side of the square inscribed in the triangle whose base is 12(b) and altitude is 6(A).

29.

in

A rectangle whose length is twice its breadth is

an equilateral

triangle.

side of the triangle is 2.

inscribed

Find the area of the rectangle

if

a

.

APPLIED MATHEMATICS

156

Find the area of a trapezoid, given the bases 36 and 56 and & 2) and the altitude 12 (A.).

30. (p 1

The

31.

bases of a trapezoid are 73 and 57

each of the nonparallel sides

One diagonal

32.

is

17

(b

Find the

(c).

and

52)

The segments

of a trapezoid is 10(d).

the other diagonal are 6 and 9 (m and

and

1

area.

of

Find the segments

n).

of the first diagonal. 33. A trapezoid contains 480 (65) sq. ft. and its altitude 20 (10) ft. Find the bases of the trapezoid if one of them 4 (6) ft. longer than the other. 34. Find the area of a rectangle if and the sides are in the ratio 3 5(ra :

The dimensions

35.

respectively.

A

Upon area

rectangle

is

its

is

50 (d)

is

ft.

diminished by 10 (m), how much in order to retain the

8(h) in breadth and

the diagonal as a base a triangle

equal to that of the rectangle.

is

diagonal

n).

of a rectangle are 64 and 58(6 and h)

If the length

must the breadth be increased 36.

:

is

is

its is

same area

diagonal

is

?

20 (d).

constructed whose

Find the

altitude of the

triangle. 37.

and

The ratio sum is

their

38.

The

rhombus is 7 5 (m Find the area of the rhombus.

of the diagonals of a 16"(/c).

:

sides of a right triangle are x, x

+ 7,

and x

:

n)

+ 8.

Find them. 39. Two telegraph poles 25 and 30 ft. high are 80 on level ground. Find the length of the wire.

40.

ment

The chord is

2(A).

of a circle

Find the

is 8(c)

and the height

ft.

of the seg-

radius.

41. In a circle whose radius is 12(r)in. a chord 4(c) drawn. Find the height of the segment. 42.

Two

chords 48 and 14


diameter of the circle

?

apart

mm.

in. is

long are on opposite sides

If they are 31

mm.

apart,

what

is

of

the

GEOMETRICAL EXERCISES 43.

Two

circle are

the circle 44.

8 (r)

157

parallel chords on the same side of the center of a 48 and 14 (14 and 4) in. long. If the diameter of is 50 (16) in., find the distance between the chords.

Find the common chord of two equal circles of radius each circle has its center on the circumference of

in. if

the other.

Two chords AB and CD intersect at E within a circle. AE = 10 (4), BE = 12 (9), and CD = 23 (12), find CE and ED.

45.

If

46.

From a

drawn.

point without a circle a secant and a tangent are

segment of the secant 18 (ri), find the tangent.

If the external

internal segment

is

is

6 (m) and the

47. If from a point without a circle two secants are drawn whose external segments are 3 and 4(3.2 and 5.5), and the internal segment of the latter is 17(4.7), what is the internal segment of the former ? 48.

The distance between the centers of two circles whose and 8(rt and r2) is 26(d). How far from the center

radii are 5

of each circle does their

(Two

centers ? 49.

The

common

radii of

two

and 4 in. The distance Find the length of the common

circles are 7 in.

between their centers is 12 in. internal 50.

tangent intersect the line of

solutions.)

and external tangents.

Find the radius of a

circle if the

numerical measure of

the area equals the measure (a) of the radius;

(ft)

of the

circumference. 51.

Find the side of the largest square piece of timber that

can be cut from a log 14 52.

A rectangle and

ft.

in circumference.

a circle

have equal perimeters. Find the is 9 (12) in.

difference of their areas if the radius of the circle '

and the width of the rectangle

is

f (£)

its

length.

53. A circle whose radius is 8(r) has one half of its area removed by cutting a ring from the outside. What is the width

of the ring ?

158

APPLIED MATHEMATICS Show

54.

that the ratio of the square inscribed in a semi-

circle to the square inscribed in the entire circle is 2

Show

55.

that the ratio of the square inscribed in a semi-

circle to the square inscribed in is

8

:

5.

:

a quadrant of the same

circle

5.

56.

What

the ratio of the square inscribed in a quadrant

is

of a circle to the square inscribed in the entire circle ? 57.

How much must be added to the circumference of a wheel

whose radius 58.

If

equator,

an

is

make the

radius 1 (m) longer ?

were laid around the earth at the would have to be added if the cable

electric cable

how many

were raised 10 59.

2 (V) to

ft.

feet

above the surface of the earth

?

A quarter-mile running track is to be laid out with straight

parallel sides

and semicircular

ends.

The

track

is

to be 10

wide, and the distance between the outer parallel edges

ft.

to

What must be the extreme length of the field so runner may cover the exact quarter of a mile by keeping

be 220 that a

is

ft.

in the center of the track ? 60. In any triangle whose sides are a, I, and c derive a formula for the square of the side opposite an acute angle. 61.

Derive a corresponding formula for the square of the

side opposite an obtuse angle. 62.

7, 8, and 9(4, 5, and and 8 (4 and 5) on 9 (6).

In a triangle whose sides are

find the projections of the sides 7

6)

63. In a triangle whose sides are 10, 12, and 18(40, 80, and 100) find the projections of the sides 18 and 10(100 and 40)

on 12(80). 64.

The

sides of a triangle are 4, 5,

and 7(70,

90,

and

100).

Find the altitude to base 7(100). 65.

Find the area of a triangle whose sides are and 30).

(20, 25,

8, 12,

and 15

GEOMETRICAL EXERCISES

159

66. Find the length of the common chord of two circles whose radii are 5 and 8 (10 and 17) and the distance between whose centers is 10(21). 67.

The base and altitude of a triangle are 8 and 6 (b and h) in. in., how much must the

respectively. If the base be increased 4 (c) .

altitude be diminished in order that the area

remain the same

?

Through the vertex of a triangle whose area is 120 in. a line is drawn dividing it into two parts, one containing 24 (12) sq. in. more than the other. What are the segments into which the base is divided if the whole base is 68.

(100) sq.

20 (14) in.? 69. The base of a triangle is 6 in. and the altitude is 5 in. Find the change in area if the dimensions are (a) increased by 3 in. and 2 in. respectively (b) diminished by 3 in. and 2 in. respectively (c) one increased by 3 in. and the other diminished by 2 in. What is the per cent of change in each case ? ;

;

At a distance of 60 ft. from a building the angles of and bottom of a tower on the building are 45° and 30° respectively. Find the height of the tower. 70.

elevation of the top

71. If the shadow of a tree is lengthened 60(a) ft. as the angle of elevation of the sun changes from 45° to 30°, how

high

is

the tree ?

A

ladder resting against a vertical wall forms an angle 72. of 60° with the level ground. If the foot of the ladder is drawn

10 ft. farther out from the wall, the angle formed with the ground is 30°. Find the length of the ladder.

and 16(20 and 40) from the vertex of the right angle to the hypotenuse, and the segments of the hypotenuse. 73.

In a right

triangle

whose

legs are 12

find the length of the perpendicular

74.

The

legs of a right triangle are 9

and 12 (a and

b).

Find

their projections on the hypotenuse. 75.

The

projections of the legs of a right triangle on the

hypotenuse are 5| and 9| (w and

ri).

Find the

legs.

APPLIED MATHEMATICS

160

The sum of the three

76.

sides of a right triangle

is

60 (140) in.

and the hypotenuse is 26 (58) in. Find the legs and the perpendicular from the vertex of the right angle to the hypotenuse. In a right triangle the perpendicular from the vertex of

77.

is 2 (j>) and the ratio of the 4:9(m:w). Find the area of

the right angle to the hypotenuse

segments of the hypotenuse

is

the triangle.

The perpendicular from the vertex

78.

of the right angle

makes the segment of the hypotenuse adjacent to the longer leg equal to the shorter leg. Find the area of the in a triangle

triangle 79.

when

the hypotenuse

is 2(c).

Two

roads cross at right angles at A. 5 mi. from A on man travels toward A at the rate of 3 mi. per hour. from A on the other road another man travels toward A

one road a 6 mi.

at the rate of 6 mi. per hour.

When and

where

will the

men

be 2 mi. apart ? 80. Two trains run at right angles to each other, one at 30 and the other at 40 mi. per hour. The first train is 15 mi. from the crossing and is moving away from it the second is 60 mi. from the crossing and moving toward it. When and where will the trains be 50 mi. apart ? ;

How much must the length of a rectangle 16 by 12 by A) be increased in order to increase the diagonal 4 (c) ?

81. (b

82.

one of

The its

difference between the diagonal of a square and sides is 2.071 (a) in.

Find one side and the

area.

83. Find the sides of a rectangle if the perimeter -is 34 (p) and the diagonal is 13 (d) in. 84.

The diagonal and longer

in.

side of a rectangle are together

5 times the shorter side, and the longer side exceeds the shorter by 7. What is the area of the rectangle ?

The perimeter

of a right triangle is 24(216) and the 24(1944). Find the sides. (Solve with one, then with two, and then with three unknowns.) 85.

area

is

GEOMETRICAL EXERCISES From

86. 6-in.

a square piece of tin a box

is

squares from the corners and folding

the volume of the box

is

864 (1944)

cu. in.,

161

formed by cutting

up the edges. If what was the size

of the original piece of tin ? 87. The sum of the volumes of two cubes is 35(2728) cu. in. and the sum of an edge of each is 5 (22) in. Find their diagonals.

88. If the edges of a rectangular box were increased by 2, 3, and 4 in. respectively, the box would become a cube and its volume would be increased by 1008 cu. in. Find the edges of

the box. 89. The diagonal of a box is 125 in., the area of the lid 4500 sq. in., and the sum of the three coterminous edges 215 in. Find the three dimensions.

90.

A rectangular piece of

is

is

cloth shrinks 5 per cent in length

and 2 per cent in width. The shrinkage of the perimeter is 38 in. and of the area 862.5 sq. in. Find the dimensions of the cloth.

a given square be subdivided into four (w 2 ) equal squares and a circle inscribed in each of these squares, the sum of the areas of these circles will equal the area of the 91. If

circle inscribed in

92.

the original square.

In a square whose side

16 a square

is

is

inscribed

by

In this square a similar manner. This is re-

ing the mid-points of the sides, in order.

another square

is

inscribed -in

peated indefinitely. Find the area of the

first

eight inscribed

squares.

In any triangle a triangle is inscribed by ing the Another triangle is inscribed in this inscribed triangle in a similar manner, and so on indefinitely. 93.

mid-points of the sides.

How

does the area of the sixth triangle compare with the area

of the first ? 94.

An


of radius 4(r).

is

circumscribed about a circle

Find a side of the

triangle.

APPLIED MATHEMATICS

162 95.

A

and 7 (a, from the 96.

whose sides are 5, 6, Find the distances of the points of

circle is inscribed in a triangle b,

and

c).

vertices of the triangles.


97.

A boy

of water

and

l

isosceles

b.^).

finds that the top of a float

How

just visible.

is

a;

+ x) = + 8000x =

(8000

x2

l

r

far does

the float project above the water

Solution,

(b

places his eyes at the surface of a smooth body

and

away

1 mi.

an

circle inscribed in

trapezoid whose bases are 6 and 18

?

2 .

l.

Since x is very small compared with the diameter of the earth, we may drop x".

= 5280 x 12 m -^oob— .

* 98.

A man

6

on the horizon.

ft. tall

How

* IG

-

-

79

standing on the seashore sees an object far, in miles, is the object

away from

the shore ? 99.

From

the top of a

funnel of a steamer, far

is

cliff

the steamer from the

100.

60

ft.

known to be 30

high

ft.

is

barely visible the

above the surface.

How

cliff ?

The bridge of a steamer is 40 ft. above the water. How two such steamers when the bridge of one is just

far apart are visible

from the bridge of the other

?

101. In a circle whose radius is 5(r) a chord 8(c) Find the length of the chord of one half the arc. 102.

Find the side of a regular polygon of twelve

is

drawn.

sides in-

scribed in a circle of radius 6(f). 103.

Find the side of a regular octagon inscribed in a

circle

of radius 8(r). 104.

The area inclosed by two concentric circles is 50 (a) sq. ft.

If the radius of the inner circle

the outer

circle.

is

5 (r)

ft.,

find the radius of

GEOMETRICAL EXERCISES 105.

163

Three men buy a grindstone. If the diameter is 3 (d) ft., of the radius must each man grind off in order to

how much

obtain his share ? 106.

The sum

of the circumferences of

and the sum of their areas

141-^ sq.

is

two circles is 56f ft. Find their radii.

ft.

(7T=V-) 107. The area of a rectangular table whose length is 5 ft. more than its breadth is equal to the area of a circular table whose radius is 3^ ft. Find the dimensions of the table.

On

a straight line 8 (m) cm. long as a diameter describe On each half of the given line as diameters describe semicircles within the other semicircle. Find the radius 108.

a semicircle. of the circle 109.

An

which

enlarges the area 110.

is

tangent to the three semicircles.

increase of 2

The

by 4

ft.

an equilateral triangle Find the side of the triangle..

in one side of

V3 sq.

ft.

sides of a rectangle are 8

and 12(6 and

h).

Find

the area of an equilateral triangle whose sides through

the vertices of the rectangle. 111. is

The number which expresses the area of a right triangle number which expresses the length of

1 greater than the

the hypotenuse. is

Show that

the

sum

of the legs of the triangle

2 greater than the hypotenuse. 112.

Find the side of the square inscribed in the common

part of two circles of radius

6(?-), if

the center of each circle

is

on the circumference of the other. 113. Two parallel lines'are 8 and 12 in. long respectively, and are 4 in. apart. Find the area of the two triangles formed by ing their opposite extremities. 114.

whose

How many

squares

sides are 9, 12,

may

be inscribed in a triangle

and 15?

In a triangle whose sides are 3, 3, and 4 (a, a, and c) a drawn across the sides 3 and 4 (b and o) bisects both the perimeter and the area. How far from the vertex does the line 115.

line

cut the sides ?

.

CHAPTER XIV VARIATION 91. Direct variation.

man

If a

earns $25 per week, the

amount he earns in a given time equals $25 multiplied by number of weeks. a

Number of weeks Amount earned

As

the

= 25 n.

.

number

the

1

2

3

4

5

25

50

75

100

125

of weeks changes the

amount earned changes,

but always the amount earned divided by the number of weeks equals 25. n.

25.

n

We may state this

fact in another

way and

say that the amount

k n.

earned varies directly as the number of weeks, or a If a steel rail weighs 100

lb.

per yard, the weight of the

equals the length in yards multiplied by 100.

w — = 100. 100,

w = 100

Since the weight divided by the length

we may

state this fact in the

form of

is

w oc

or

constant,

variation,

that the weight varies directly as the length, or

rail

I,

and say

I.

Note that in direct variation an increase in one variable makes an increase in the other. The greater the length the greater the weight

;

the less the length the less the weight.

Double the length and the weight

is

doubled

the length gives one fourth the weight. 164

;

one fourth of

VARIATION One number

92. Definition.

the quotient of the

first

165

varies directly as another

divided by the second

is

when

constant.

Exercise. On a sheet of squared paper take the lines at the bottom and left for the axes of x and y respectively, and let one square each way equal one. Draw a straight line from the lower left corner to the intersection of any two heavy lines.

Make a

table for the values of x, y,

and — for points on

this

line, taking x = 1, 2, 3, Is the quotient of y divided •, 10. by x constant ? Does y vary directly as x ? What equation connects y and x ? •

PROBLEMS 1.

The weight

volume.

.

If

weigh 7.5

150

of a mass of brass varies directly as

cu. in.

weigh 45

how many

lb.,

its

cubic inches

lb. ?

Solution. Given

wxv.

(1)

By

-

=

(2)

definition,

w = kv. = k 150. k = .3.

(3)

45

Substitute values,

Solving for

k.

k,

(4) (5)

= .3 v. v = 25 cu. in.

Substitute in (3),

7.5

(6)

Arithmetical solution.

The weight

1 cu. in.=

of

.3 lb.

loU 7L5 - -

Hence it requires 2.

—45- =

=

25 cu.

in. to

weigh

7.5 lb.

Construct a graph to show the relation between the volis the equation of the

ume and weight in Problem 1. What straight line ? Eead off some sets of

values from the graph

and check by the equation. 3.

The weight

ume. If 60

cu. in.

of a

mass of gold varies directly as its vollb., find the weight of 35 cu. in.

weighs 42

APPLIED MATHEMATICS

166

Construct a graph to show the relation between the

4.

vol-

ume and weight of a mass of gold on the same axes as in Problem 2. What does the difference in the slope of the two graphs show?

The

5.

distance through which a body falls from rest varies

as the square of the time during

400

ft.

in 5 sec,

how

far will

which

it fall

it falls.

If a

body

falls

in 20 sec. ?

Suggestion, d oc t 2 Check by arithmetical solution. 20 -s- 5 = 4. Since the distance varies as the square of the time, the body will fall .

400 x 4 2 6.

=

6400

ft.

in 20 sec.

Construct a graph to show the relation between distance

and time in the case of a falling body.

A. man wishes

93. Inverse variation.

to lay out a flower

bed containing 120 sq. ft. If he makes it 12 ft. long, it must be 10 ft. wide 20 ft. long, 6 ft. wide and so on. The greater the ;

;

length the less the width.

If the length

doubled, the width

is

120

is

halved

;

always

lb

= 120, or = —— We say that the length I

.

varies inversely as the width,

94.

when

Definition.

and write

One number

varies

1

.

it I oc

-

inversely

another

as

their product equals a constant.

Suspend a meter stick at its center so as to baland attach a 500-g. weight 6 cm. from the fulcrum. Suspend on the other side a 100-g. weight to balance. How far from the fulcrum is it ? Suspend other weights to balance, and make a table for the weights and distances from the fulcrum. Multiply each weight by its distance from the fulcrum. What seems to be true ? If w d = 3000 (a constant), we may say Exercise 1.

ance,

that the distance varies inversely as the weight, dec Exercise 2.

in Exercise 1,

—

-

Locate on squared paper the points from the table and draw a curve through them. Express the

relation between x

and y

(1) as a variation

;

(2) as

an equation.

VAKIATION

167

PROBLEMS

The time it takes to do some work varies inversely as the number of men at work. If 6 men can do the work in 10 da., how long will it take 5 men to do it ? 1.

Solution. Let

APPLIED MATHEMATICS

168

PROBLEMS 1. The volume of a cylinder varies tly as the altitude and the square of the radius of the base. When the altitude is 20 in. and the radius of the base is 10 in., the volume is 6284 cu. in. Find the volume when the altitude is 8 in. and

the radius of the base

Solution.

is

6

v

6284 * v

2.

in.

v oc ~hr\

The pressure

of

= khr2 = £-20-10 2 = 3.142. = 3.142 x 8 x 6 2 = 904.9 cu. in. .

.

wind on a plane surface

varies tly as

the area of the surface and the square of the velocity of the If the pressure on 100 sq. ft. is 125 lb. when the wind blowing 16 mi. per hour, what will be the pressure on a plateglass window 10 by 12 ft. when the velocity of the wind is 70 mi. per hour ?

wind. is

97. Suggestions for the solution of problems in variation. 1.

From

the

conditions given in the problem write the

variation. 2.

Change the variation

3.

Substitute the given numbers and find the value of the

constant 4.

to an equation.

k.

In the equation substitute the value of k and the other

numbers given

in the problem!

5.

Solve this equation for the required number.

6.

Check.

While most of the problems in the following list should be solved by the principles of variation, some of them may be solved more easily by proportion. All results should be checked, and as far as possible the meaning of the constant should be discussed.

VARIATION

169

PROBLEMS

The circumference. of a circle varies directly as its diamand when the diameter is 17.5 in. the circumference is 55.0 in. Find the circumference when the diameter is 22.7 in. 1.

eter,

The

2.

velocity acquired

by a

falling

body varies as the 4 sec. is 128.8 ft.

If the velocity acquired in

time of falling.

per second, what velocity will be gained in 7

sec. ?

3. The weight of a mass of gold varies directly as its volume. If 5 ccm. weighs 96.3 g., how many cubic centimeters will weigh 1kg. ?

The area

4.

square of

its

of the surface of a cube varies directly as the

What

edge.

will be the

edge of a cube the area

whose surface is 315| sq. in., if the area of the surface of a cube whose edge is 3^ in. is 73^ sq. in. ? of

The simple

5.

time during which

sum

is

on a sum of money varies as the If the interest on a certain what will be the interest for 8£ yr. ?

interest it

$84.20 for 5

bears interest.

yr.,

6.

The

7.

If the friction

working load on a rope varies as the square of girth. If the safe load on a manila rope 6 in. in girth is T., find the girth of a rope whose safe load is 3.6 T.

its

1.2

safe

between a wagon and the roadway varies on the wheels, and if the friction is 241b. 650 lb., find the friction when the load is l£ T.

as the total load

when

the load

8.

is

The distance a body

under -the action of gravity If a body falls in 5 sec, in how many seconds will it fall 680 ft. ? falls

varies as the square of the time of falling.

403

ft.

9.

The surface

of a sphere varies as the square of

If the surface of a sphere

radius of 7

in.

is

sq. in.,

its

radius.

by how much must

be increased in order to double

its

its

surface ?

Given that the extension of a spring varies as the stretchis stretched 10 in. by a weight of lb., what weight will stretch the spring 7.5 in. ?

10.

ing force, and that a spring 5.2

616

APPLIED MATHEMATICS

170 11.

The

safe load on a rectangular

beam

the breadth and the square of the depth. joist of

12.

by 4 in. pine what weight will the same material and length safely

given length s safely 320

a 2^ by 10 ?

in.

beam

The weight

of

lb.,

of a disk of copper cut

form thickness varies

varies tly as

If a 2

from a sheet of uniFind the

as the square of the radius.

weight of a circular piece of copper 12 7 in. in diameter weighs 4.42 oz.

in.

in diameter,

if

one

13. The volume of a quantity of gas varies as the absolute temperature when the pressure is constant. If a quantity of gas occupies 3.25 cu. ft. when the temperature is 14° C, what

will be its

volume

at 56.5° C. ?

(Absolute temperature

= 273° + the reading of the Centigrade

thermometer.)

volume of a certain gas is 376 ccm. when the 12° C, at what temperature will the volume he the pressure remaining the same ? ccm., 533.3 14.

If the

temperature

is

15. Find the volume of a gas 200 ccm. at 27° C. 16.

at

— 23° C,

if its

volume

is

If the quantity of water that flows through a circular

pipe varies as the square of the diameter of the pipe, and if 1.02 gal. per minute flow through a half-inch pipe, how many gallons per minute will flow through a 3-in. pipe ? 17. The safe load on a wrought-iron chain varies as the square of the diameter of the section of the metal forming a link. If the safe load on a chain in which the metal is £ in.

thick

is

900

lb.,

what diameter of metal

will be necessary in a

chain that is to bear a load of 6.4 T. ? 18.

What

is

the safe load for a chain in which the diameter

of a section of the metal forming a link

is .9 in. ?

19. The quantity of heat generated by an electric current in a given conductor for a given time varies as the square of the

VARIATION number

of amperes.

171

Find the amount of heat generated by a if 224 units of heat are generated by

current of 25 amperes,

a current of 16 amperes. 20.

A

current

is

found to generate 350 units of heat in the How many amperes in the current ?

conductor of Problem 19. 21.

The compression of a spring under a given load varies mean diameter of the coils, other conditions

as the cube of the

being the same. is

1.64

in.

22.

The

When

What is

the diameter

the compression

deflection

is

4

in.

the compression

when the diameter is 6£ in.

ed at both ends varies as the cube of

beam 9

ft.

long

is

23.

The diagonal

5

13.4

what

in.,

length.

A

ft.

of a cube varies directly as the edge of is 8.66 in. when its edge edge of a cube whose diagonal is

If the diagonal of a cube

the cube. is

its

by a certain load. Find the long by the same load.

deflected .135 in.

beam 15

deflection of a

?

by a given load at the middle of a beam

will be the

in. ?

24.

A

solid sphere of radius 3.5 in.

weighs 12

lb.

What

is

the diameter of a sphere of the same material that weighs

given that the weight of a sphere varies as the cube of

96

lb.,

its

radius ? 25.

The

distance in miles of the offing at sea varies as the

square root of the height in feet of the eye above the sea level. If the distance

distance

when

is

4 mi. when the height

the height

is

121.5

is

10

ft.

8

in.,

find the

ft.

According to Boyle's law the volume of a gas varies when the temperature is constant. If the volume of a gas is 600 ccm. when the pressure is 60 g. per square centimeter, find the pressure when the volume is 150 ccm. 26.

inversely as the pressure

27. If the

12.6

lb.

35.7 cu.

volume of a gas

is

42.5 cu.

per square inch, find the pressure in.

in.

at a pressure of

when

the volume

is

APPLIED MATHEMATICS

172

The pressure allowed

28.

versely as

pressure allowed

when

allowed

is

104

in a cylindrical boiler varies in-

When

diameter.

its

lb.

the diameter

Equal quantities of

the diameter

per square inch.

96

is

42

is

What

the

in.

pressure

is

in. ?

on opposite sides of a piston of moves 4 in. from the center, find the ratio of the pressures on the two sides of the piston. 29.

a cylinder 16

The

30.

in. long.

air are

If the piston

intensity of light varies inversely as the square of

the distance from the source of light. gas jet at a distance of 10

50

at

ft.

31.

when latter.

32.

as

18

A

ft. is /,

If the illumination of a

what

will

it

be at 20

ft. ?

?

student lamp and a gas jet illuminate a screen equally placed 12 ft. from the former and 20 ft. from the

it is

Compare the

relative intensities of the

two

lights.

How far from

much

a lamp is a point that receives three times light as another point 20 ft. away ?

33.

How much

farther from a gas jet

in.

away from

it,

two thirds

much

as

must a book, which

be removed in order that

it

may

is

receive

light ?

An

8 candle power electric lamp at a distance of 6 ft. from a screen illuminates it with one half the intensity of a candle at a distance of 1 ft. 6 in. from the screen. What is the 34.

candle power of the candle ? 35.

In a given latitude the time of vibration of a pendulum its length. If a pendulum 39.1 in.

varies as the square root of

long vibrates once in a second, what is the length of a pendulum that vibrates twice in a second ? three times in a second ? 36.

The

velocity with

which a liquid flows from an

orifice

varies as the square root of the head (depth of the liquid above

the

A

orifice).

reservoir 40

when an opening

is

made

water escapes with an

What would height of 8

ft.

ft.

is

initial velocity

be the velocity ?

high

filled

with water, and

in the side at a height of 4

if

of 48

ft.

ft.,

the

per second.

the opening weTe

made

at a

VARIATION 37.

The weight of a body

173

varies inversely as the square of

distance from the center of the earth.

If a body weighs 100 lb. at the earth's surface (4000 mi. from the center), what would be its weight at the summit of the highest mountain, which is 5£ mi. high ? its

38. How far above the earth's surface must a body that weighs 150 lb. at the surface be removed, in order that its weight may be reduced to 96 lb. ? 39.

The diameter

square root of plate,

what

its

of the rivets used for a plate varies as the If l|-in. rivets are used for a 1-in.

thickness.

size of rivets is required for

a

f-in. plate ?

What

thickness of plate can be riveted with |-in. rivets ? 40.

The volume

of a gas varies inversely as the height of

the mercury in a barometer, the temperature being constant.

mass occupies 32 cu. in. when the barometer reads what space will it occupy when the reading is 30.4 in. ?

If a certain

28.8

in.,

41. Compare the amounts of heat received at two points whose distances from the source of heat are in the ratio 4 3, assuming that the intensity of heat varies inversely as the square of the distance from the source of heat. :

'

magnet for a piece of iron varies and if the distance of .1 in. is a, what will be the

42. If the attraction of a

inversely as the square of the distance between them, the attraction at

attraction at .2 in. ? at .3 in. ? at .5 in. ? 43.

The

attractive force

between two oppositely

electrified

balls varies inversely as the square of the distance

them.

At

a distance of 8 cm. the force

distance will the force be .64 44.

The compression

is

3.5 g.

between

At what

g. ?

of a spring under a given load varies

inversely as the fourth power of the diameter of a cross section of the steel in the coils, other conditions being the same.

the compression

is

be the compression

3.5 in.

when

the diameter

when thediameter

is

1\

is

f in.,

in. ?

what

If will

APPLIED MATHEMATICS

174 45. If

7

men

in 9

weeks earn $516.60, how many men

take to earn $360.80 in 4 weeks,

it

amount earned varies number of weeks ? 46.

The volume

tly as the

5 cm.

radius.

its

and 3 cm., and and recast

respectively, are melted thick. 47.

What

is its

4f

in.

48.

its

in.

4.95 com.

its thick-

12 cm. and 20 cm.

into a single disk 6 cm.

varies tly as

lb.,

what

is

its

in.

length

long and

the diameter of a

lb. ?

of a cone varies tly as

the square of the radius of is

radii

long that weighs 135

The volume

the

metallic disks having

If a cylinder 12

diameter.

in diameter weighs 49

cylinder 20

men and

of

radius ?

The weight of a metal cylinder

and the square of

number

Two

will

being given that the

of a circular disk varies tly as

ness and the square of thicknesses

it

when its

base.

its

If the

its altitude and volume of a cone

is 2.1 cm. and its radius is 1.5 cm., whose radius is 3 cm. and volume

altitude

find the altitude of a cone

33 ccm. 49.

How

far

from a light of 9 candle power

will the illumi-

nation be 2\ times the illumination at a distance of 24 a light of 16 candle power ? 50.

The weight

ft.

from

of a uniform bar of given material varies

and the area of its cross section. If a steel and 1 ft. long weighs 3.3 lb., what is the weight of a T-rail 2 ft. long and 8| in. in cross-sectional area ? tly as its length

bar 1

51.

sq. in. in cross section

An ohm

is

the resistance offered to the flow of an electric

current through a column of mercury 106 cm. long and 1 in cross-sectional area.

What

mercury 3 m. long and 4

sq.

is

sq.

mm.

the resistance of a column of

mm.

in cross-sectional area, the

resistance varying directly as the length and inversely as the cross-sectional area ? 52.

A

wire of diameter .0704

in.

has a resistance of 15 ohms.

Find the diameter of a wire of the same length and whose resistance is 5.4 ohms.

material

VARIATION 53. If the resistance of

175

500 yd. of a certain cable

is

.65

ohm,

be the resistance of 1 mi. of a cable of the same material and of one half the cross-sectional area ?

what

'will

54.

Find the resistance of 1000 yd. of copper wire

.15 in. in

diameter, if the resistance of 112 yd. of copper wire .06 in. in

diameter

1 ohm.

is

Solve also by the formula in Exercise 10,

page 73, and compare the results. nitions 55.

The

resistance of a certain wire

resistance of

2£ mi. of the same wire

Find the length of the 56.

XVII

See Chapter

for defi-

and explanations.

The

is

1.82 ohms, and the

known

to be 3.25 ohms.

first wire.

resistance of

cross section 11.2 sq.

is

mm.

2400 is

ft.

of a certain copper wire of

1.13 ohms.

What

is

of 2 mi. of copper wire of cross section 6.45 sq.

the resistance

mm.

?

According to Ohm's law the number of amperes flowing through an electric circuit varies directly as the number of volts 57.

of electromotive force

and inversely

as the

number of ohms

such as to main10 amperes through a resistance of 40 ohms, what would be the current if the electromotive force were doubled and the resistance diminished by one third ? If the voltage in a certain circuit

resistance.

is

tain a current of

58.

How many

amperes are there in the current maintained

by a dynamo whose resistance is 2.4 ohms, that of the rest of the circuit being 17.6 ohms, and the electromotive force 210 volts ? 59.

The

resistance offered

by the

air to the

age of a

bullet through it varies tly as the square of its diameter

and the square of its velocity. If the resistance to a bullet whose diameter is .32 in. and whose velocity is 1562.5 ft. per second is 67.5 oz., what will be the resistance to a bullet whose diameter is .5 in. and whose velocity is 1300 ft. per second ? 60.

From

the>

data of Problem 59 determine the diameter

of a bullet that has a resistance of 50

900

ft.

per second.

oz.

when

its

velocity

is

1

APPLIED MATHEMATICS

176

61. If t denotes the time of revolution of a planet in its orbit about the sun, and d the mean distance of the planet from the s Assuming that the earth's period of sun, then f varies as d revolution is 365 da. and that of Venus 225 da., find the ratio .

of the 62.

mean distances of these two planets from the sun. The horse power that a solid steel shaft can transmit

safely varies tly as

its

speed in revolutions per minute

and the cube of its diameter. A 120 r. p. m. can transmit 240 h.'p. be transmitted

100

r.

63.

p.

4-in. solid steel shaft

How many

the diameter of the shaft

if

is

3

making

horse power can in.

and

its

speed

m. ?

The pressure

of the

wind on a plane surface varies and the square of the wind's

tly as the area of the surface

on a square yard is 12£ lb. when the 17£ mi. per hour, what is the pressure on a square foot when the velocity of the wind is 45 mi. per hour ?

velocity.

If the pressure

velocity of the

wind

is

64. The space s ed over and the time of flight t of a body projected vertically upward are connected by the rela2 where a is constant. If s = 676 ft. when tion s = at — 16 ,

t

=

6^ sec, find

s

when

t

= 3 sec.

CHAPTER XV EXERCISES IN SOLID GEOMETRY Use short methods

98.

of multiplication

and division and

keep the results to a reasonable number of significant I.

1.

A

line 8

figures.

Numerical Exercises

long makes with a plane an angle of 45°.

ft.

Find the length of the projection of the

line

upon the

2.

What

will

3.

Prove

that, if a line is inclined to a plane at

plane.

be the length of the projection of the line in the preceding exercise, if it makes an angle of 30° with the plane? 60°, its projection 4.

*In a

ceiling

is

upon the plane

is

swimming tank the water 11

ft.

above the water

;

an angle of

equal to half the

5^

is

ft.

a pole 22

line.

deep and the ft.

long rests

obliquely on the bottom of the tank and touches the ceiling.

How much

of the pole

P

is

above the water

?

from a plane a perpendicular PQ is drawn to the plane; with Q as a center and a radius of 4^ in. a circle is described in the plane at any point R of this circle a tangent R T 10 in. long is drawn. Pind the distance from P to T. 5.

Prom

a point

6

in.

;

6. With a 12-ft. pole marked in mine the foot of the perpendicular the ceiling of a room 9 ft. high ?

feet

how can you

let fall to

deter-

the floor from

is 20 cm. from each of the vertices of a right whose legs are 12 cm. and 16 cm. respectively, find the distance from the point to the plane of the triangle. 7.

If a point

triangle

177

APPLIED MATHEMATICS

178 8.

Determine the relation between

diagonal of a face of a cube

;

(b)

(a) the edge

and the

the edge and a diagonal of

a cube.

The sum of the squares of the three edges of a rectangular is 2166 and the three edges are to each other as 1:2:3. Find the edges. 9.

parallelepiped

10. The dimensions of a rectangular bin are 4 ft., 4£ ft., and 10 ft., and it is desired to treble its capacity. How can this be done if only one dimension is changed ? two dimensions ? all three dimensions ? 11.

xa

Make a geometrical application of the equation (x

+ 3x*y + 3xf + f. 12.

How much

will it cost, at

an open ditch 80 rd. long, 6 the bottom, and 3 ft. deep ?

ft."

+ yf =

40 cents per cubic yard, to dig wide at the top, 2\ ft. wide at

13. How many square feet of lead will be required to line a rectangular cistern 10 ft. long, 7 ft. wide, and 4^ ft. deep ? What will be the weight of the lead if it is -fa in. thick and a cubic inch weighs .411 lb. ? 14.

What

the weight of the water received upon an acre

is

of ground during a storm in

an inch 15.

which rain

falls to

the depth of

?

Allowing 30

cu.

ft.

of air per minute for each person' in

how much

air must be driven into the room and how many times must the air be changed during the recitation period to insure good ventilation ?

this classroom,

ft. long is an equilateral poured into the trough, water are triangle. When whose edges are in the same horizontal plane, how deep will

16.

The

cross section of a trough 12

20

gal. of

10

ft.

the water be ? 17.

than

A its

room width.

find the other

is

high and

its

length

If the area of the ceiling

two dimensions.

is

one half greater

and walls

is

816

sq.

ft.,

EXERCISES IN SOLID GEOMETRY 18.

4

ft.

A

block of ice 1^

long and 2

ft.

by 2

What

wide.

ft.

ft.

by 3

179

placed in a box depth of water in gravity of ice being

ft. is

will be the

the box after the ice melts, the specific

.917? 19.

How

large a cubical reservoir will be required to hold

the water that falls on the roof of a house covering 548 sq. ft. of ground, during a shower in which § of an inch of rain falls ? 20.

How many

make a

tent 10

ft.

square yards of canvas will be required to

by 16

ft., if

the sides are 6

ft.

high and the

roof has £ pitch ? 21. An oblique prism whose altitude is h has for its base a rhombus whose diagonals are k and I. Find its volume.

22.

5

:

18.

Two

rectangular parallelepipeds are to each other as

The dimensions

of the first are 5, 13 £,

and

18.

Find

the dimensions of the other, if they are to each other as

1:2:3. 23. The base of a prism whose altitude is 15 cm. is a quadrilateral whose sides are 10 cm., 18 cm., 12 cm., and 16 cm., the last two forming a right angle. Find its volume. 24.

A

prism has for its base a triangle whose sides are to 12 13. If its altitude is 4 m. and its volume

each other as 5 is

:

:

4.8 cu. m., find the sides of the base. 25.

high. 26.

144

The Great Pyramid is 762 ft. square at the base and 484ft. Compute its volume and its lateral area. The

sq. in.,

lateral area of

and one

if its specific

a regular square pyramid of wood

side of the base

gravity

is

8

in.

Find

its

is

weight,

is .53.

Determine the volume of a pyramid, one of whose lateral faces is an equilateral triangle on a side of 18 in., and whose third lateral edge is perpendicular to the other two and is 24 in. long. 27.

28.

96 is

sq.

4

ft.

A

pyramid parallel to the base contains and its distance from the base whose area is 120 sq. ft. Find the altitude of the pyramid.

ft.,

section of a

APPLIED MATHEMATICS

180 29.

64

sq.

The

area of a regular triangular pyramid

lateral

and one side of the base

ft.

is

8

ft.

Find the

is

altitude.

a section of a pyramid parallel to the base is so taken is £ that of the base, what part of the pyramid is that portion above the section ? 30. If

that

its

area

31. If the sides of the base of a

and the

by a plane of the base, what

solid is cut

section is

^

pyramid are

4, 6, 7,

and

9,

parallel to the base so that the will be the lengths of the sides

of the section ? 32. A granite obelisk in the form of a frustum of a regular quadrangular pyramid, surmounted by a pyramid of slant height 15 in., has each side of one base 1 ft. 4 in. and each side of the other base 2 ft. 3 in., and the slant height is 12 ft. If the specific gravity of the granite is 2.6, find the weight of the

obelisk. 33.

What

will be the expense of polishing the faces of the

obelisk in the preceding exercise at 50 cents per square foot ? 34.

What

is

depth and 300

the capacity in gallons of a reservoir 12 ft.

A

ft.

wide at the

ft.

in

top, the slope of

3:2?

the walls being 35.

long by 160

granite

monument

in the form of a prismoid

high and the dimensions of 18 in. by 12 in. respectively.

is 16 ft. ends are 42 in. by 28 in. and What is its weight if the specific

its

See Kent's " Mechanical Engineers' Pocket-Book" for the definition of a prismoid and for gravity of the granite

is

2.7 ?

the prismoid formula. 36.

A

milldam of earth with plane sloping sides and recis 80 m. by 6 m. at the top and 66 m. by 18 m.

tangular bases at the bottom. 37.

If its height

How many

is

5.4 m., find its cubic contents.

cubic yards of earth will

it

be necessary to

remove in making a cut for a railroad, which must be 14 ft. deep, 24 ft. wide, 240 ft. long at the bottom, and 170 ft. long at the top, the slope of the sides being 7

:

10

?


Apply the prismoid formula

whose edge

181

to the regular octahedron

is e.

39. The volume of a wedge whose base is 7.5 cm. by 12 cm., and whose height is 3.5 cm., is 142 ccm. Find the length of its

edge, regarding the solid as a prismoid. 40.

3

in.

Find the weight of a steel wedge whose base measures by 7 in., the edge 5 in., and the height 6 in., if a cubic

inch of steel weighs .283

lb.

weighs 490 lb., what is the weight square at one end, 7 in. at the long, the metal being | in. thick ?

41. If a cubic foot of steel

of a hollow steel

beam 10

other end, and 18

ft.

in.

Find the cost of painting the lateral surface of an octagonal tower whose slant height is 40 ft., if the short diameter of the lower base is 12 ft. and of the upper base 3 ft., at 24 42.

cents per square yard. 43.

At what distance from the vertex of any pyramid must ft. long be cut by planes parallel to the base,

a lateral edge 12

in order that the areas of the sections

other as 2 44.

3

:

:

5

formed may be to each

?

What must be sum of

the height of a prism of iron equal in

weight to the

three other prisms of iron of the same

shape, the height of the latter being 2

in.,

3

in.,

and 4

in. re-

spectively ? 45.

What

A

block of granite weighs 2 T. and

its

width

is

3

ft.

the width of a block of granite of the same shape whose weight is 6 T. ? 46.

is

A block

and a block of

of

wood

of specific gravity .675 weighs 72.4

steel of specific gravity 7.84

shape weighs 13.14

lb.

Find the

lb.,

and of the same

ratio of their corresponding

dimensions. 47.

Of two bodies

its specific

gravity

specific gravity is 2.32.

60 cm., what

is

same form, one weighs 2 lb. and while the other weighs 56 lb. and its If one dimension of the first body is

of the

is .24,

the corresponding dimension of the second ?

APPLIED MATHEMATICS

182

An

48.

42^

mass of

irregular

a mass of gold of the same form, two corresponding lines of the two

specific gravity 19.3, if

masses have the ratio

How much

49.

iron, specific gravity 7.2, weighs

What is the weight of

lb.

2:3? make an open

tin will be required to

which

drical vessel of altitude 65 cm.

cylin-

shall contain 160.2

1.,

taking no of seams ?

What

50.

from a is

the amount in cubic feet of evaporation daily

is

circular fishpond 6 rd. in diameter, if the loss in depth

.04 in. ?

How many board

51.

from a round log 20

The

52.

4

ft.

in.

feet of

two

areas of

lumber 16 in. wide can be made and 16 ft. long ?

in diameter

sections of a cylinder of revolution

high, which are parallel to the axis

to each other, are

ft.

sections are 2

6 sq.

and and 4§ sq. ft. respectively. If the what is the volume of the cylinder ?

apart,

53. If a cubic foot of copper is

diameter, what will be

An

54.

irregular

its

drawn

into a wire -^

in.

in. in

length ?

mass weighing 21.07

kg.

is

dropped

into a

cylindrical vessel 42 cm. in diameter, partially filled with water.

If the water rises 80 cm., find the volume

and

specific gravity

of the body.

How many

55.

cubic yards of stone will be required for a

semicircular culvert under a railroad bank 112

throat of the culvert being 6 56.

in diameter,

and 1

57.

A

58.

What

in. thick.

is its

wide, the ft.

thick ?

14 ft. 4 in. long, 6 in. weight if the specific

7.2 ?

steel shaft is

in. to 4.5 in.

weight

is

ft.

high and the walls 2

A hollow cylindrical iron column is

gravity of iron

5

ft.

Find

reduced in diameter in a lathe from

to the nearest

hundredth what part

of

its

is lost.

In what time

of water of 1

ft.

will a 1-in. circular pipe in

per second

cistern of dimensions

3£

ft.

is

by 4

maintained, ft.

by 7£

ft.

fill

?

which a flow a rectangular

EXERCISES IN SOLID GEOMETRY The plunger of a

59.

certain single-acting

pump

183 is

10

in.

in

and makes 15 strokes per minute. many gallons of water through it in 12 hr. ?

diameter, has a 10-in. stroke,

How

A

60.

pump

Holley

of water.

per minute, what

When

61.

water in 24

make

has an hourly capacity of 145,800 gal.

If the plunger has a 40-in. stroke

if its

a

hr.,

is its

diameter

and makes 18 strokes

?

pump is required to furnish 2,800,000 gal. of how many strokes per minute must the plunger

diameter

is

30

in.

and

stroke

its

is

40

in. ?

The

following rule is sometimes used to calculate the horse power steam boiler. To the heating surface afforded by the flues is to be added two thirds of the lateral surface of the boiler, and two thirds of one flue sheet diminished by the ends of the flues. In general practice 12 sq. ft. of heating surface are considered to afford of a

1 h. p.

Compute the horse power of a steam boiler whose length and diameter 6 ft., if there are 136 flues, each 16 ft. long and 3 in. in interior diameter. 62.

is

16

ft.

What must

63.

of diameter 2

be the length of the flues of a steam boiler

containing 34 2-in. flues, in order that

ft.,

it

may

afford 12 h. p. ? 64.

A conical heap of grain 4 ft. high covers a space

diameter on the

hold 65.

it

floor.

How

12 ft. must be a cubical bin

large

to

?

How many

a conical tent 10

square yards of canvas are required to

ft.

high, such that a

without stooping anywhere within 4 66.

in

A conical vessel whose

angle

man

ft.

is

6

ft. tall

may

make stand

of the center ?

60°

is filled to

the depth

and when a solid cube of wood is submerged the water rises 1 in. Find the edge of the cube.

of 8 in. with water, in

it,

67.

How much

height, 68.

ground is covered by a conical tent 9 which contains 162 sq. ft. of canvas ?

A square whose side

diagonals.

ft.

is 4 cm. revolves around one of Find the volume generated.

in

its

APPLIED MATHEMATICS

184 69.

A

in. by 8 in. revolves around one of its Determine the volume and the area of the surface

rectangle 6

diagonals.

generated. 70. If

28

ft.

a sector of 120°

that can be 71.

A

made is

iron weighs .261

The

cut out of a circular piece of canvas

out of the remainder

hollow iron cone

and the metal 72.

is

what are the dimensions of the

in diameter,

\

4

is

weight

its

in.

if

in diameter,

a cubic inch of

lb.

altitudes of a cylinder

each other as 16

long and 4

in.

Find

in. thick.

conical tent

?

:

27.

Find the

and an equivalent cone -

are to

ratio of their other dimensions.

73. At 15 cents per square foot, what will be the cost of cementing the walls and bottom of a cistern in the form of an inverted frustum of a cone of revolution whose depth is 7 ft. and diameters 6 ft. and 3 ft., the lid \\ ft. square not being cemented ? 74.

A cone whose

slant height

three parts in the ratio of 1

from the vertex, must the

:

2

16 cm.

is

is

At what

3.

:

to be divided into

distance, measuring

slant height be cut

by

planes parallel

to the base ? 75. In a sphere of radius 5 ft., what whose plane is 4 ft. from the center ? 76.

In a sphere of radius 6

77.

is

50f

radius 5

ft.

On

a small

whose plane

is

4£

ft.

the area of the

far

circle

from the center

is

sq. ft. ?

of 120° in the circumference

What is the length of an arc

of a circle

78.

how

ft.

the plane of a circle whose area

is

from the center of a sphere

of

?

a sphere of radius 6

circle

whose latitude

in.

what

60°

is

is

the polar distance of

What

?

is

the radius of the

circle ? 79.

How many

spherical triangle 80.

degrees

in

whose area

is

each angle of an equilateral -fa

If a birectangular triangle

sphere,

what

is

of that of the sphere ? is

^

of the surface of

the third angle of the triangle ?

its

EXERCISES IN SOLID GEOMETRY 81. If the

moon is 2162 mi., find volume in cubic miles.

diameter of the

in square miles

and

its

The diameter of the earth

82.

is

185 its

surface

7918 mi. and that of the

planet Mercury 3030 mi.

If the density of the latter

times that of the former,

show

that the

is

2.23

mass of Mercury

is

nearly £ that of the earth. If the mean diameters of the earth and the moon are 2162 mi. respectively, show that the ratio of their and 7918 surfaces is 27 2 nearly. 83.

:

What

84.

is

the diameter of a sphere of which a wedge of

11° 15' contains 359.3 cu. dm. ?

How many bullets

85.

of lead pipe

A

86.

1\

in.

1

steel ball 6-in. in

vessel 8 in. in diameter,

with water.

How much

made

in diameter can be

in.

in exterior diameter

diameter

which

is

and \

in.

of 3

ft.

thick ?

dropped into a cylindrical

within 2 water will overflow ? is filled

in.

of the top

87. If 400 lead balls each \ in. in diameter are melted and run into a disk T^ in. thick, what will be the radius of

the disk?

How many bullets

88.

be

of caliber .32 (.32

made from a bar

of lead 2\ in.

A marble

in diameter

by 4

in.

in. in

diameter) can

by 6 in.

?

dropped into a conical glass f whose diameter is 2 in. and depth 3 in., and is just covered by the water that it contains. What was the depth of the water 89.

in.

is

at first?

What

90.

is

the altitude of that zone of a sphere which

equals a trirectangular triangle in area ? 91.

cut off 92.

Find the surface of the zone of a sphere of radius 8 by a plane 3 in. from the center of the sphere.

What

is

the altitude of a zone of 120 sq.

the radius of the sphere 93.

How

far

is

10

in.

in.

surface, if

in. ?

from the surface of a sphere must a lamp be may be illuminated?

placed in order that one sixth of the surface

APPLIED MATHEMATICS

186

Show

94.

visible to

that the portion of the earth's surface

an aeronaut at a height h above the surface

is

r

When h

r being the radius of the earth.

is

small

+h

it

may

be dropped in the denominator, giving the approximate area 2 irrh. 95. On a globe of radius 7 cm. it is desired to mark off a zone whose area shall be 6.16 sq. cm. What opening of the comes shall be used ? 96. On a globe of radius 9 in. a small circle is described with an opening of the comes of 6 in. Find the length of

the circumference. 97.

are 12

The altitude and radius of the base of a right cone and 9 in. respectively. Find the radius of the circle of

tangency of the inscribed sphere. 98.

How

does the specific gravity of a spherical body com-

pare with that of a liquid in which

it floats,

surface above the surface of the liquid just

with one half

one third

?

when

its

it is

submerged ?

99. If a .3

?

sphere of oak 6

in. in

diameter

floats in

water with

of its surface above the surface of the water, what

specific gravity of the

100.

oak

is

the

?

What portion of the surface

of a ball of iron of diameter

and specific gravity 7.2 will remain visible when it is dropped into a dish of mercury whose specific gravity is 13.6 ? 1 in.

II.

99.

A

Graphical Exercises

few of these exercises should be worked out

carefully

in the notebook. 1. Construct a graph to show the change in the volume of a cube as its edge increases from to 12 in. What is the equation of the graph ?


On

the same axes as in Exercise 1

change in the surface of the cube. («)

when the surface

How

its

equation. Y

show graphically the do the graphs show

equals the volume numerically ? (b)

a cube has a greater surface than volume

?

187

when

Write on each graph

APPLIED MATHEMATICS

188

The

4.

the base

altitude of a right cone

is

9

Show

ft.

is

12

ft.

and the radius

graphically (a) the volume,

eral surface of the cones cut off

the

(b)

of

lat-

from the vertex by planes

parallel to the base. 5.

On

the same axes construct graphs to show the change

(a) in volume, (b) in lateral surface of a right cylinder the radius

of

whose base 6.

6

is

in.,

as its altitude increases from

to 15

Represent graphically the change in the area of a

in.

section

parallel to the base of a regular triangular pyramid, the side

of

whose base 7.

volume, to

is

8 cm. and whose altitude

On the same

10

(b) in

12 cm.

is

axes represent graphically the change

(a) in

surface of a sphere as the radius increases from

in.

8. The volume of a pyramid is 60 cu. in. Construct a graph show the relation between the base and altitude as the altitude to 180 in. increases from

to

to

9. The volume of a cylinder is 440 cu. in. Construct a curve show the relation between the radius of the base and the

altitude, as the radius increases 10.

From each

from

to

10

in.

corner of a square piece of tin 12

side a smaller square

is cut,

in.

on a

the remainder of the sheet being

bent so as to form a rectangular open box. Determine the side of each small square in order that the capacity of the box may be as great as possible. 11.

If the sheet of tin in the preceding exercise had been

rectangular,

20

in.

by 12

in.,

what then would have been

the

size of each small square ? 12.

A

bin with a square base and open at the top

constructed to contain 400 cu.

ft.

dimensions to require the least 13.

A

gal.

steel plate in the

What

is

What must

amount of material

to be

be

its

?

oil tank is required to hold 100 bbl, dimensions will necessitate the least

closed cylindrical

each of 42

of grain.

making

?

EXERCISES IN SOLID GEOMETRY

189

An

open rectangular tank whose length is to be twice to hold 200 gal. of water. What dimensions will require the least amount of lining for the tank ? 14.

width

its

15.

is

The strength

of a rectangular

the product of its breadth

are the dimensions of the strongest

a round log 2 16.

beam

is

and the square of

beam

proportional to

its

What

depth.

that can be cut from

in diameter ?

ft.

If the slant height of a right cone

be the radius of

its

base in order that

12 ft., what must volume may be as

is

its

great as possible ? 17.

Determine the right cylinder of greatest

that can be inscribed in a cone of revolution

14

in.

and radius of base 8

lateral surface

whose

altitude

is

in.

Find the dimensions of the smallest cone of revolution whose altitude and radius are respectively 9 dm. and 3 dm. 18.

that can be circumscribed about a cylinder

19.

of

its

The

the stiffest 20.

stiffness of a rectangular

breadth and the cube of

its

beam

varies as the product

depth. Find the dimensions of

beam that can be sawed from a log 20

in.

in diameter.

Determine the dimensions of the largest right cone that

can be inscribed in a sphere of radius 5

in.

Find what radius of the base of a conical tent of 375 cu. ft. capacity will require the least amount of canvas in the making. Also find the relation between the altitude and the radius. 21.

22.

Find the radius of the right cylinder of greatest

surface that can be inscribed in a sphere

whose diameter

lateral

is

12

in.

Find the relation between the radius of the base and the altitude of a right cone whose convex surface contains 264 sq. ft., in order that the volume may be as great as possible. 23.

24.

Determine the altitude of the least cone of revolution dm.

that can be circumscribed about a sphere of radius 2 25.

What must be

the altitude of the cone of revolution of

least lateral surface that

whose radius

is

4

in. ?

can be circumscribed about a sphere

APPLIED MATHEMATICS

190

Algebraic Problems

III.

Make a

100.

1.

What

Put the given dimenup the equations from the sketch.

sketch for each problem.

and

sions on the figure

set

are the other two dimensions of a rectangular

parallelepiped whose length

and

its total

2.

surface

is

184

is

8

in., if its

volume

160

is

cu. in.

sq. in. ?

If the three face diagonals of a rectangular solid are

respectively

and 9

6, 7,

cm.,

what must be the dimensions

of

the solid ? 3.

One dimension

one diagonal faces

is

is

44

is

12

of a rectangular parallelepiped

6

is

in.,

and the area of one of the wholly unknown What are the other two dimensions ?

in.,

sq. in.

4. The sum of the three dimensions of a rectangular solid 12 and the diagonal of the solid is 5 Vz. Find its total surface'. 5.

The sum

of a diagonal and an edge of a cube

End

is 6.

an edge of the cube. 6.

The area

that of another

of one face of a rectangular solid

15

is

sq. cm.,

and the

is

total area is

10 sq. cm., 100 sq. cm.

Find the dimensions. 7.

What

are the dimensions of a rectangular solid whose

entire surface

one end 40

is

392

sq. in., if its top contains

96

sq. in.

and

sq. in. ?

Given the diagonal of a cube equal to k. Find the volume of the cube and its surface. 8.

9.

Given the volume v and the altitude h of a regular hexagFind s, the length of one side of the base.

onal prism. 10.

and

The

its

sides of the base of a triangular prism are as 3

volume

is

432

cu. ft.

If the altitude

is

4

ft.,

:

4

:

5,

find the

sides of the base.

What must be the altitude of a pyramid in order that total area may be equal to the sum of the areas of two similar

11. its

pyramids whose altitudes are respectively 6 and 4

in. ?

EXERCISES IN SOLID GEOMETRY What

12.

98

sq. in., if

the altitude of a pyramid whose base contains a section parallel to the base and 4 in. from the

is

vertex contains 32 sq.

in. ?

pyramid with a rectangular base is 76.8 is 9.6 in., and the altitude exceeds of the base by 2 in. Find the altitude and the

The volume

13.

191

of a

one side of the base

cu. in.,

the other side

other side of the base. 14. If

a square pyramid has each basal edge equal to

each lateral edge equal to «

e

v show that the volume

e

and

will be

= |V2(2e *-*). 1

15.

Given

v,

the volume, and

s,

one side of the square base

of a regular quadrangular pyramid, find the lateral surface. 16.

Derive an expression for the volume of a regular tetra-

hedron in of

its

An iron plate

edge

e.

long and 2\ in. thick has squared ends but uniformly and equally beveled sides, and contains 122 cu. in. 17.

8

in.

If the difference of the widths of the

two

flat faces is 2.8 in.,

find those widths. 18.

The

pyramid

is

lateral area of

281.2 sq.

in.,

a frustum of a regular quadrangular

the slant height

is

15.2

in.,

of the lower base exceeds a side of the upper base

and a side by 3.75 in.

Find a side of each base. 19.

which of

its

20.

What must is

diameter

?

The sum of the numerical measures

lateral area of is

be the diameter of a cylindrical gas holder

to hold 6,000,000 feet of gas, if its height is to be %

a cylinder of revolution

14,

what

21.

Write the formula that gives

is

is

of the

231.

volume and

If the altitude

the diameter ? t,

the total surface of a

cylinder of revolution, in of h, the altitude, radius of the base,

and solve

for

it

h and

In case of a cylinder of revolution 22.

Given

t

and

r,

find

h and

v.

:

r.

and

r,

the

APPLIED MATHEMATICS

192 23.

Given v and

24.

Given v and

h, find r

25.

Given

26.

27.

r,

h and

find

t.

and

I

(the lateral area).

I

and

v,

r,

and

£.

Given

I

and

A, find r, v,

and

£.

Given

t

and

i>,

and

Z.

Suggestion.

Find

find h,

find

by

r-

r, h,

trial

from 2

numerical values of t and v (see and then I from Z = 2 ?rrti.

sect.

tit"

—

58)

;

fr + 2 u = for any given then find h from v = irr\

How far

from the axis of a cylinder of revolution whose and diameter d ft. must a plane parallel to the axis be ed, in order to make a section of area k sq. ft. ? 28.

height

is

h

ft.

cone of revolution

29. If the total surface of a

the slant height 30.

The sum

is 4,

of the altitude

cone of revolution

is

volume of the cone

?

The

31.

tude

is

is

21

-w

and

and the volume of the cone.

find the radius

and the radius

11 and their product

lateral area of a right cone

equal to 3 times the radius of

is

its

is

of the base of a 10.

What

9 VTOtt, and

base.

Find

its

is

the

its alti-

volume.

Find the slant height and the radius of the base of a whose total surface is 462 sq. in., and the sum of the slant height and the radius is 21 in. 32.

cone of revolution

33.

The

34.

What

right cone,

77

7r,

The

.

is

the radius of the upper base of a frustum of a

is

volume

if its

and the radius of 35.

whose slant height Find the radius of the base.

lateral surface of a right cone

5 exceeds the base by 12f

its

.516

is

lower base

lateral area of a

the slant height

is 7,

it cu. .8

dm.,

dm.

its

altitude 1.2 dm.,

?

frustum of a cone of revolution is altitude is 2 V<3. Find the

and the

radii of the bases. 36.

What

is

the volume of a frustum of a right cone the sum

of the radii of

whose bases

altitude being 7 ?

is

11 and their product

28, the

EXERCISES IN SOLID GEOMETRY Find the

37.

radii of the bases of a

given the lateral area as 1068f sq. and the altitude as 15 ft.

ft.,

193

frustum of a right cone,

the slant height as 17

ft.,

38. The volumes of two spheres are to each other as 8 125, and the sum of their radii is 12 in. Find the radii. :

The product

39.

of the radii of two spheres

ratio of their surfaces is

25

:

What

64.

is

22.5 and the

are the radii ?

a sphere is equal to the sum of the two spheres whose radii are 2 in. and 4 in. respechow does its volume compare with the sum of their

40. If the surface of

surfaces of tively,

volumes

24

?

What

41.

sq. in.

is

is

the radius of a sphere of which a zone of

illuminated

by

a

lamp placed 18

in.

from

its

surface ?

What

42.

relation

must the radius of a given sphere bear

the radii of two other spheres if its surface

between their surfaces

tional 43.

is

a

mean

to

propor-

?

Compare the expression for the volume of a sphere with and determine how long the radius must order that the volume may be numerically greater than

that for its surface,

be in

the surface. 44.

In a sphere of radius 8 the radius of one small circle

is

mean proportional between the radius of the sphere and the radius of another small circle, and the sum of the radii of the

a

two small 45.

circles is 10.

Find the

radii of the small circles.

Derive an expression in one variable for the volume of

a right cone inscribed in a sphere of radius

r.

Find an expression in of the altitude for the total surface of a cylinder of revolution inscribed in a sphere of 46.

radius 47.

r.

What is

the expression for the volume of a right cylinder

inscribed in a right cone, altitude h, radius of base of the radius of the cylinder ?

r,

in

APPLIED MATHEMATICS

194 48.

Find an expression in one variable

for the total surface

of a right cone circumscribed about a given right cylinder. 49.

What

expression in one variable denotes the volume of

a right cone circumscribed about a given sphere

?

Derive the expression in one variable for the surface of the cone in Exercise 49. 50.

51.

Find an expression in one variable

for the

lateral

volume

of a

right cone circumscribed about a given right cylinder.

Problems 45-51 furnish good exercises in maxima and minima

by giving numerical values to the dimensions of the constant solids. Since some of the expressions are rather complicated the work of computing the table of values may be divided among the of the class, each one computing the value of the function for a single value of the variable.

CHAPTER XVI HEAT Thermometers. Though the Fahrenheit scale is in and in ordinary engineering work, the Centigrade scale is used in laboratories and all scientific work to such an extent that one should become acquainted with 101.

general use in everyday life

Fahrenheit (Danzig, ) devised his scale about

it.

1726.

He

thought that the lowest possible degree of cold was

obtained by mixing salt and ice

;

hence he took as zero the

mercury when placed in such a mixture. It is not known why he marked the boiling point of water 212°. The Centigrade scale was proposed by Anders Celsius (Upsala,

position of the

Sweden) about 1741. In the Fahrenheit thermometer the boiling point of water at is taken at 212° and the freezing point of water at 32°.

sea level

In the Centigrade thermometer the boiling point is taken at 100° and the freezing point at ? Hence 180° on the Fahrenheit .

scale equals 100°

on the Centigrade

scale.

= 100° C. 1°F. = |°C. 1° C. = |° F.

180° F.

It

(1) (2)

should be ed that a division on the Centigrade

than a division on the Fahrenheit scale. Hence from degrees Centigrade to degrees Fahrenheit we

scale is longer

in changing

get a greater

grade

we

number

and from Fahrenheit number of degrees.

of degrees,

get a smaller

to Centi-

Equations (1) and (2) enable us to change readily from one scale to the other.

195

APPLIED MATHEMATICS

196

PROBLEMS 1.

Construct a graph to change a number of degrees of one Why is it necessary to

scale to degrees of the other scale.

locate only 2.

F.

;

two points and draw a straight

(d) 80° C. to F.

;

line

through them

?

200° F. to C. (c) 40° C. to 150° F. to C. (/) 112° F. to C. Check

(a) 90° F. to C.

Change

(e)

;

(b)

;

;

by the graph. 3.

The sum

degrees C. C.

is

of a

number

When

121.

of degrees F.

and a number of

the degrees F. are changed to degrees

and added to the number of degrees number of degrees F. and C.

C. the result is 85.

Find

the

4. The sum of a number number of degrees C. is 53.

degrees

sum is

is

73.

102.

and a number of

of degrees F. If each

changed into the other scale, the Find the number of degrees F. and C.

To change thermometer readings from

one scale to the other.

In the above problems

we were

dealing with degrees not with ther-

mometer

readings.

eter readings

When we

from one

change thermom-

scale to the other

we

must take of the difference in position of the zeros on the two scales. Thus find the C. reading when the F. reading Looking at Fig. 81, we see that by takis 80°. ing 32° from 80° we get 48°, the number of degrees the F. reading is above 0°C. Then 48° F.

= 48°x

|C.

=

26.7°C.

Similarly, to find the F. reading

reading

is

when

the C.

70°,

70° C.

=

70°

x

f F.

= 126° F.

Fir 81

But 126° F. takes us only to 32° F. opposite 0° C. Hence we add 32° to get the F. reading, 158°, corresponding to 70° C

:

HEAT To change from F.

to C.

197

readings, subtract 32°

the difference by |.

To change from

by | and add 32°

the product.

to

C.

F.

103.

and multiply

C. to F. readings, multiply

= |(F. -32°). = §C. + 32°.

To determine the relation of the corresponding readtwo thermometers by experiment. Take several

ings of the

readings of the two thermometers on different days, or obtain the readings by putting the thermometers into water at different temperatures.

Readings obtained

CO

32°

F.

47°

70°

96°

118°

151°

8

21

36

48

66

Locate these points on squared paper. Units 1 large square

= 10°

F., vertical,

;

C,

horizontal,

1 large square

= 10°. On

stretching a thread along these points

they

lie

nearly in a straight

line.

Draw

it

:

will be

found that

a straight line

among

the points so that they are distributed evenly above and below it.

This line

is

the graph of the equation which connects the

corresponding readings. that

it is

of the

form

To _,

F.

find the equation

we

will suppose

„

= fflO. + 6,

(1)

where a and b are unknown numbers which must be determined. Taking the second and fifth points and substituting the readings in (1), we have

^=

118 Solving these equations,

Sa +

= 48 a + we

get a

b

b.

= 1.77, b = =

32.8.

+

1.77° C. 32.8°. Substituting these values in (1), we get F. The readings in the experiment were not taken with sufficient

exactness to give a close result (see

sect. 59).

Exercise. Take several corresponding readings on the two thermometers and find the relation as above.

:

APPLIED MATHEMATICS

198

PROBLEMS Construct a graph to change the readings of one thermometer to those of the other. Units horizontal, 1 large Take the vertical, 1 large square = 10° C. square = 20° F. 1.

:

;

lower left-hand corner as the origin and mark it — 40°. Show that — 40° is the same reading on both scales. Locate one other point.

Why

is

the graph a straight line

Change the reading of one thermometer other, and check by the graph: 2.

(e) 195° F. toC.

(a) 78° F. to C.

3.

?

to that of the

-20°F. toC.

(b)

18°F.toC.

(/)

(c)

88°C.toF.

O) -

(d)

60°C.toF.

(h) 0° F. to C.

30° C. to F.

The melting point of the following metals Change to the Centigrade scale

is

given in

degrees F.

Tin Lead

... .

.

.

Silver

.

.

.

Gold

.

.

.

442° to 608° to

446°

Copper

618°

Cast iron

1733° to 1873° 1913° to 2282°

4. The following record The Chicago Daily News.

Steel

.

.

.

Platinum

of temperature

1929° to 1996° 1922° to 2075°

2372° to 2532° 3227°

was taken from

HEAT Change the readings

199

to Centigrade

by using the graph, and

on the same sheet of squared paper plot a curve for each of the two sets of readings. 5.

What

temperature

Are the curves is

Why ?

parallel ?

expressed by the same number on

the F. and C. scales ?

A

Fahrenheit and a Centigrade thermometer are placed and the F. reading is found to be double the C. reading. What is the temperature of the liquid in degrees C. ? 6.

in a liquid

Expansion of Solids by Heat 104. Linear

expansion.

changes of temperature

its

When

a

solid

length changes

subjected to

is ;

in general, the

rises, and decreases as For ordinary temperatures the amount of change

length increases as the temperature

it

falls.

is

nearly the same for each degree of increase or decrease. following table gives results that have been secured

ment

;

by

The

experi-

they are only approximate.

Linear Expansion of Solids for 0° and 212° F.

1

Degree, between

Aluminum

.00001234

Lead

.00001571

Brass, plate

.00001052

Platinum Steel, cast

00000479 00000636

Copper Glass, white Iron, wrought

.

.00000887

.

.00000478

Steel,

.00000648

Tin

Iron, cast

.

.00000556

Zinc

.

.

.

tempered

.

.

.00000689

00001163 00001407

The amount of expansion is seen to be very small. Thus when we say that the linear expansion of wrought iron is .00000648 we mean that the length of a wrought-iron rod 100 ft. long increases 648 millionths of a foot when the temperature of the rod rises 1 degree.

made a

However, provision must be work for example,

for this expansion in all construction

little

space

is

left

between the ends of the

;

rails in

laying

railway track, hot-water pipes have telescopic ts, and so on.

APPLIED MATHEMATICS

200

PROBLEMS 1.

Find the linear expansion of copper, wrought

and

iron,

tin for 1° C. 2.

A brass wire

A

200

long at

ft.

0°.

Find

its

length at 85°.

200 x .00001052 x 85 = .179 ft. 200 + .179 = 200.179 ft. = 200

Solution.

3.

is

66

steel chain is

ft.

long at 77°.

What

ft.

2.2 in.

will be its

length at 32°? 4. The iron girders of a railway bridge are 100 ft. long at a temperature of 10°. What will be the length of the girders at 90° ? 5.

at

A

lead pipe

is

80

ft.

long at

— 10°. How long

will it be

What

length at

100°? 6.

A

brass rod

is

5 m. long at 0° C.

is its

38° C. ? 7.

is

What

is

the length of a wrought-iron rod at 0° C.

1,56 m. long at 100° C. 8.

What

length 1.2 9.

in.

What

is

the length of a copper wire which increases in

when is

its

temperature

is

raised 200° ?

the area of a brass plate at 100° C. which

measures 8.35 cm. by 4.16 cm. at 0° C. 10.

What 11.

One brass yardstick is

if it

?

is

?

correct at 32° and another at 68°.

the difference in their lengths at the same temperature ?

A bar of metal 10ft. long at 200° increases in length .31 in.

when heated

to 362°.

Find the expansion of 1

ft.

for 1°.

12. A plate-glass window is 10 ft. by 12 ft. How much will change in area when its temperature changes from — 20° to 90°, if its linear expansion for 1° is .000005 ?

it

13.

An

iron steam pipe 200

perature from 190° to

— 4°.

ft.

long at 190° ranges in tem-

What must

be the range of motion

of an expansion t to provide for the change in length ?

HEAT 14.

A

platinum wire and a brass wire are each 100 ft. long must they be heated to make the brass wire longer than the platinum wire ?

How much

at 30°.

1

in.

201

Let

Suggestion.

x

=

the

.00001052 x 100 x 12 x 15.

A copper bar

is

10

number

-

ft.

of degrees.

.00000479 x 100 x 12 x x

long.

What must be the may expand

a cast-iron bar in order that the two

amount

=

1.

length of the same

for 1° ?

A steel tape 100 ft. long is correct at 32°. On a day when temperature was 88° a line was measured and found to be

16. its

What was

1 mi. long.

the error and what was the true length

of the line ? 17.

An- iron casting shrinks about \ in. per linear foot when down to 70°. What is the shrinkage per cubic foot ?

cooling

18. The Chicago and Oak Park Elevated Railway is about 9 mi. long from Wabash Avenue to Oak Park Station. What is

the difference in the length of the rails for a change in 20° to 80° ?

temperature from 19.

100

ft.

20.

Construct a graph to show the expansion of a steel wire

long as

The

water as graph.

-

its

temperature rises from 0° to 2000°.

following table shows the change in the volume of temperature rises from 0° to 17° C. Construct the

its

How

does the graph show an exception to the law that

the volume increases with a rise in temperature ?

Temp.

;

APPLIED MATHEMATICS

202

Measurement op Heat

When

105. Units.

a definite quantity of heat

is

applied to

various substances different effects are produced, depending on

An amount of heat which can be measured but it has been found convenient to measure heat by considering the change in temperature it produces. the nature and condition of the substance.

may

be expressed by any of

Two (B.

t.

unit

heat units in general use are the British thermal unit

and the

u.) is

its effects

For ordinary engineering work the

calorie.

the British thermal unit, the amount of heat required

to raise 1

of water 1° F.

lb.

investigations

is

A

smaller unit used in laboratory

amount of heat required to The amount of heat required to raise

the calorie, the

raise 1 g. of water 1° C.

a quantity of water 1 degree varies with the temperature the variation

is

so small that in practical

and say that the same amount of heat will raise 1 water from 10° to 11° or from 211° to 212°. lect it

106.

The

relation between heat

wood, boring

iron,

becomes heat.

and

It has

;

but

work we may neglb.

of

and work. In sawing work

so on, a part of the energy of

been found possible to determine the

work required to raise the temperanumber of degrees. The famous experiments of Joule, in England, in the years 18431850, showed that 772 ft. lb. of work raised the temperature of

number

of foot pounds of

ture of a quantity of water a certain

water at 60°

F.,

1 degree.

His apparatus consisted of a brass cylinder in which water was churned by a brass paddle wheel, which was made to revolve by a falling weight. Later experiments by other methods have given results more nearly exact, and by general consent it is now considered that 778 ft. lb. of work are required to raise the temperature of 1 lb. of water 1° F.

1 B.

1

t.

u.

ft. lb.

= 778 ft. lb. = .00129 B.

t.

u.

HEAT

203

PROBLEMS

How many

1.

British thermal units are required to raise

the temperature of 120

the boiling point

lb.

of water from the freezing point to

?

On a cold day in winter a tank 1 ft. by 2 ft. by 8 ft. was with water at a temperature of 100°. When the water had reached the freezing point, how much heat had been given out ? 2.

filled

If 1 lb. of coal contains 13,500 B.

3.

t.

of heat,

u.

how many

pounds of coal would be required to raise the temperature of 12 cu. ft. of water 50° if there was an efficiency of 10 per cent ? 4.

Find the number of British thermal units required to 20 lb. of lead from 70° to the melting

raise the temperature of

point, 608°.

lead 1° as 5.

heat

(It takes

it

given

lb.

of steel 1° as

How many foot pounds

temperature of 20

8.

lb.

it

to raise 1.1b. of

of

(It takes

only .12 as

does to raise 1

lb.

much

of water 1°.)

work are required

to raise the

of water 12° ?

778 x 20 x 12

Solution. 7.

70° ?

off as it cools to

heat to raise 1

to 70°.

much heat

of water 1°.)

lb.

A steel ingot weighing 1 T. is red-hot (1200°). How much

is

6.

only .03 as

does to raise 1

The temperature

How many foot

=

186,720

ft. lb.

ft. of water was raised from 32° pounds of work did it require ?

of 1 cu.

Through how many feet would a weight of 1200 lb. have enough energy to raise the temperature of

to fall to generate

8

lb.

of water 15° ?

9. Find the distance through which a weight of 2 T. could be raised by the expenditure of an amount of heat that would raise the temperature of 2 lb. of water 30°.

10.

How many

perature

of.

Solution.

10

horse power would it take to raise the temwater from 70° to 120° in 1 hr. ?

cu. ft. of 62.4

——

x 10 x 778 x 50 rr-rrr 60 x 33,000

= ,„„ 12.3

,

h. p. r

;

APPLIED MATHEMATICS

204 11.

Find the number of British thermal units per minute

required for an engine of the following dimensions diameter of cylinder, 50 in. stroke, 36 in. revolutions per minute, 54 :

;

;

mean effective pressure, 28 lb. per square inch. Find also the number of pounds of coal required per hour, if 1 lb. of coal contains 13,500 B.

t.

and only 10 per cent of the heat

u.

of the

coal reaches the piston.

How many calories

12.

of 40 13.

g.

are required to raise the temperature

of water 20° C. ?

If

126 calories of heat raised the temperature of a how many grams of water were there ?

quantity of water 49° C,

The temperature

14.

of 1

1.

of water

How many

to the boiling point.

was raised from 40° C. were required ?

calories

15.

How many

16.

Construct a graph to change calories to British thermal

calories are there in a British thermal unit ?

units.

Specific 107. Exercise. similar dishes.

on an

Heat

Put equal weights of water and mercury in Note the temperature of each. Place the dishes

After a time found that there is a considerable difference in the temperatures of the mercury and water. Since the mercury and the water have received the same it

electric stove or in a dish of boiling water.

will be

amount

of heat, it is evident that it takes less heat to raise the temperature of 1 lb. of mercury 1° than is required for 1 lb. of water. It is found by experiment that equal weights of different substances require different amounts of heat to raise their

temperatures the same number of degrees. requires 1 B.

t.

requires .2 B. 1

lb.

t.

u. to raise its u.

;

of ice requires

1

lb.

.5

B.

Thus

temperature 1° F.

;

1

lb.

of water

1

lb.

of glass

of cast iron requires .12 B. t.

t.

u.

;

and

u.

The specific heat of a substance is the quoby dividing the amount of heat required to raise

108. Definition. tient obtained

the temperature of a given weight of

it

1°

by the amount of

;

HEAT

205

heat required to raise the temperature of an equal amount of water 1°.

The

(Note the similarity to specific density.)

specific heat of substances varies a little

perature, but in practice

it is

with the tem-

considered to be constant.

Table of Specific Heat Aluminum

0.21

Silver

Brass

0.09

Steel

Copper

0.09

Tin

Glass

0.20

Zinc

Iron, cast

0.12

Water

0.11

Ice

0.03

Steam

0.03

Air

Iron,

wrought

Lead Mercury

.

.

.

.

.

.

.

0.06

.

.

.

.

0.12

0.06

.

.

.

...

.

.

.

0.09

.

1.00

.

0.50

.

0.48

....

0.24

PROBLEMS

How many

British thermal units are required to raise 1. the temperature of 10 lb. of copper from 70° to 200° ? 200° - 70° = 130°. Solution. would require 1300 B. t. u. to raise the temperature of 10 water 130°. Specific heat of copper = .09. 1300 x .09 = 117B.t.u. It

lb.

of

.-.

2.

How many

calories are required to raise the temperature

of 500 g. of lead 40° C. ?

500 x 40 x

Solution.

.03

=

600

calories.

Find the amount of heat required to raise the temperalb. of silver from 70° to the melting point, 733° 30 lb. of aluminum from 70° to the melting point, 1157° 25 lb. of ice from 0° to 32° (d) 1 kg. of mercury 80° C.

3.

ture of (a) 20 (l>)

(c)

;

4.

How many

5.

If 1

thermal units are given off by an iron casting which weighs 50 lb., as it cools from 2000° to 70° ? lb.

British,

of water at 70°

mercury at 70° are how hot will the mercury

and 1

given the same amount of heat, become when the water is at 73° ?

lb.

of

APPLIED MATHEMATICS

206

Equal weights of tin and cast iron are put into a tank When the tin has been heated 10°, how much has the iron been heated ? 7. If 15 lb. of water at 200° and 8 lb. of water at 70° are 6.

of boiling water.

mixed together, what

the resulting temperature ?

is

Solution. Let

t

15 (200

—

= the

resulting temperature.

= number

of British thermal units lost.

— 70) = number of British thermal units gained. 8 (t- 70) = 15 (200-08

(t

= 154.8°. = 154.8) 8 (154.8 x 45.2 = 8 x 84.8. 678 = 678.4.

Solving,

Cheek.

t

15 (200 15

70).

8. 20 lb. of water at the freezing point are poured into 25 lb. water at the boiling point. What is the temperature of the of mixture ?

9.

A

tank 2

ft.

by 3

If the tank

at 60°.

ft.

by 6

is filled

ft. is

two thirds

water

full of

with water at 120°, what

is

the

temperature of the mixture ? 10.

60

lb.

How many

pounds of water at 40° must be mixed with 100° to obtain a temperature of 80° ? of water at

p = number

Solution. Let

of

pounds at

40°.

— 40) = number of British thermal units gained. 60 (100 — 80) = number of British thermal units lost. p (80 - 40) = 60 (100 - 80). p (80

Check.

11.

1 cu.

30 (80 60 (100

p = 30 lb. - 40) = 1200 B. t. u. gained. - 80) = 1200 B. t. u. lost.

How many ft.

pounds of water

at 180°

must be mixed with

of water at 56° to obtain a temperature of 112° ?

12.

How many

13.

How much

grams of water at 0° C. must be mixed with 100° water at C. to obtain a temperature of 80° C. ? 1 kg. of

must be taken

to

water at 212° and

make up 36 lb.

how much water

at 97° ?

at 32°

HEAT — number of pounds = number of pounds x + y = 36. z(212- 97) = ^(97-32). 115 x = 65 y. * = i*yin (1), \% y + y = 36. V = 23. in (1), x = 13. 13 (212 97) = 23 (97 - 32). 13 x 115 = 23 x 65. 115 = 115.

Solution. Let

Substitute (4) Substitute (6) Check.

14.

207

How much

x

at 212°.

y

at 32°. (1) (2) (3) (*)

(5) (6) (7)

water at 180° and at 81° must be taken to

a tank which contains 90 lb., if temperature of the mixture 125° ? fill

it is

desired to have the

Into a dish containing some water at 4° C. was poured some water at 75° C. How many grams of each were taken if there were in all 250 g. at a temperature of 60° C. ? 15.

16.

An iron casting when

red-hot (1300°) was put into a tank

If the temperature of the water rose to 200°, what was the weight of the casting ?

containing 2 cu.

ft.

of water at 170°.

Solution. Let

x

Specific heat of cast iron

2 cu.

(1300

—

ft.

of water

200) x .12 x

= number of pounds of cast iron. = .12. = 62.4 x 2 = 124.8 lb. of water. = number of British thermal units lost by the

(200

(1300

Check.

-

170) x 124.8

- 200)

28.4

x

iron.

= number

of British thermal gained by the water.

units

= (200 - 170) x 124.8. = 3744. = 28.4. .12 = 3749 B. t. u. lost. 30 = 3744 B. t. u. gained.

.12 x 132 x x

x 1100 x 124.8 x

mass of lead at 500° was put in a gallon of water and the temperature of the water rose from 77° to 80°, what was the weight of the lead ? 17.

(8£

If a

lb.)

APPLIED MATHEMATICS

208 18.

An

80-lb.

mass of

many pounds 19.

1 cu.

If 20 ft.

lb.

steel at

1000°

is

placed in a tank con-

If the final temperature

taining water at 60°.

is

70°,

how

of water are in the tank ? of brass at 300° were placed in a tank containing

of water at 72°,

what would be the

final

temperature ?

20. If 500 g. of brass at 100° C. were placed in 188 g. of water at 17.5° C. and the final temperature was 33.5° C, find the specific heat of the brass.

Solution. Let 500 (100

s -

188(33.5

33.5) -

s

17.5)

= the specific heat of the brass. = number of calories lost by the brass. = number of calories gained by the water.

500(100

-

33.5)

s

S

Check.

21.

500 x 66.5 x .0905 188 x 16

=

188(33.5 - 17.5). 188 x 16

~ 500 x 66.5 = .0905. = 3010 calories = 3008 calories

gained.

The following data were obtained by experiment. Find

the specific heat of each metal.

No.

lost.

HEAT cent of the fuel

many pounds

is

209

available for raising the temperature,

of hard coal (1

lb.

coal

= 13,500 B.

t.

u.)

how

would

be required ?

Latent Heat 109. Exercise.

Place a dish of melting ice on a stove.

Though

the melting ice and water receive heat continuously, a thermometer placed in the dish will stand at 32° F. till all the ice is

melted.

reached. is

Then the mercury will rise till the boiling point is The temperature will remain at 212° till all the water

evaporated.

110. Latent heat. This heat which goes into a substance and produces a change in form rather than an increase in

temperature

is

called latent (hidden) heat.

The following table

gives the approximate

thermal units absorbed by 1

from

lb.

solid to liquid or liquid to solid.

Latent Heat of Fusion

number

of British

of the substance in changing

APPLIED MATHEMATICS

210 3.

How much

melting point

if its

Solution.

is required to melt 16 442° ?

heat is

=

Specific heat of tin

442°

- 70° =

16 x 372 x .06 16 x 27

4.

How much

.06.

= 357 = 432

B.t.u. to raise to 442°.

B.

t.

u. to melt. total.

required to melt 150 622° ?

heat

melting point

is

is

of tin at 70°

372°.

789 B.t.u.,

if its

lb.

lb.

of lead at 70°

1 T. of molten iron at 2200° cooled to 70°.

5.

heat was given

How much

the melting point was 2000° ?

off if

6. A cake of ice weighing 50 lb. is at 0°. How many British thermal units are required to melt it and bring the water to

the boiling point ? If 1 lb. of ice at 32°

7.

much

is

put into 2

lb.

of water at 80°,

how

of the ice will melt ?

Solution. (80° - 32°) 2 142.6

= 96 B. t. u. available to melt the ice. = number of British thermal units quired to melt 1

96

=

.67 lb. ice melted.

= =

80.

lb.

re-

of ice at 32°.

142.6 Check.

142.6 x .67

^8.

How much

+

32

96.

boiling water will be required to melt 12

lb.

of ice at 32° ? 9.

What would be the

temperature of the water

final

of ice at 32° were put into 40

Solution. Let

t

142.6 x 16

lb.

(t

ice.

= number of British thermal units lost. — 32) + 142.6 x 16 = number of British thermal units gained. 40 (212 - t) = 16 (t - 32) + 142.6 x 16. 40 (212

16

16 lb.

= the final temperature. = number of British thermal units to melt the

—

if

of boiling water ?

Solve and check.

HEAT 10.

5

into 50

211

of molten lead at the melting point 610° were poured

lb.

What is

of water at 70°.

lb.

11. 1 lb. of lead at

212°

?

placed on a cake of ice at 30°.

is

How much ice will it melt ? 12. How many pounds of

the resulting temperature

steam at 212° will melt 20

lb.

of

must be added

to

ice at 32° ? 13.

100

How many

lb.

14. 2.00°.

pounds of zinc

of water at 75° to heat

20

lb.

What

of ice at 0° are is

it

at 500°

to 100° ?

immersed in 200

the temperature of the water

lb.

of water at

when

the ice has

just melted ? 15.

How many

at 212°

by 1

pounds of water at 70° would be evaporated assuming an efficiency of 12 per cent,

T. of coal,

t. u. per pound The temperature of 1

and 13,500 B. 16.

from 32°

of coal lb.

?

of water in a teakettle rises

(a) How long before the dry ? (b) If the kettle contained 5 lb. of water, British thermal units would be needed to boil it dry ?

to 212° in ten minutes,

kettle will boil

how many

Solution, (a)

212°

-

32°

10 965.7

=

180°.

=

18 B.

=

_„ _ 53.7 minutes.

u.

t.

.

per minute. F ,

18 17. If 1 lb. of ice at 0° is

put on an electric stove which gives

per minute, find the number of British thermal units and the number of minutes required (a) to raise the ice to 32°; (b) to melt the ice; (c) to raise the water to 212°; out 8 B.

t.

u.

(e) to raise the steam to 312°. Conhorizontal, the results on it. Units and write graph struct a 1 large square = 10 minutes vertical, 1 large square = 20°.

(d) to evaporate the water

;

:

;

CHAPTER XVII ELECTRICITY 111. Exercise.

Into a tumbler two thirds full of water pour

2 com. of sulphuric acid.

Stand in this solution a strip of zinc

and a strip of copper each well sandpapered. Take 6 ft. of No. 20 insulated copper wire and wind about 25 turns around a large lead pencil, leaving about a foot uncoiled at each end.

Cut the insulation from the

ends of the wire and wrap the ends

around the strips, as shown in Fig. 82. To get good connections it may be necessary to cut into the edge of the strips

and wedge

under the pieces

-the wire

lifted.

Take a piece of

soft

wrought iron and

sprinkle some iron filings on each end.

Place the iron within the

coil,

shown in Fig. 82, and drop some filings on the ends. Result ? Is the

iron

ftesult ?

as

iron

Fig. 82

magnetized ? If so, we have generated a current of electricity and magnetized the iron. (See Shepardson's "Electrical Catechism.") 112. is

Nature of

not known.

ether.

ever,

electricity.

Some

Others think that

much

is

it is

known about

methods of applying

The exact nature

scientists think

it

it is

of electricity

a condition of the

a form of energy or force.

How-

the laws of electricity and about

to useful work.

113. Electromotive force.

When

the strips of copper and

zinc were placed in the solution of sulphuric acid, the acid dis-

solved the zinc strip faster than 212

it

did the copper

strip.

We

ELECTRICITY

213

say that this caused an electrical flow from the zinc to the that is, an electromotive force exists between the two Whatever produces or tends to produce an electrical flow is called an electromotive force (e. m. f.). When the two strips are connected by the wire this action takes place continuously and there is said to be a flow of electricity from the zinc to the copper and through the wire to the zinc again. Though we cannot perceive this flow by any of our senses, we

copper

;

strips.

can see the effects 114.

The

it

produces. It is not possible nor is

electrical units.

it

neces-

However, definitions can be given which are readily understood and are sufficiently exact sary to give exact definitions here.

for practical purposes.

The

volt.

between the

We may

above, as pressure. tricity

think of the electromotive force existing

strips of zinc

and copper

in the cell described

It takes pressure to force

through a wire, just as

of water through a pipe.

it

a current of

elec-

takes pressure to drive a stream

To measure

this pressure

we have

the unit of electromotive force called a volt (from Volta, an Italian physicist

who

is

moved

is

about 1.1

across the magnetic lines of force

A

When

volts.

is

volt is the electromotive force set

produced in the

up

in a wire that

crosses magnetic lines of force at the rate of one lion per second.

In a dynamo the armature

as a bundle of wires

which cut across the

a wire

which exist between

the poles of a magnet, an electrical flow wire.

The pressure

lived from 1745 to 1827).

of a gravity or crowfoot cell

hundred mil-

may be

thought of

lines of force of the

magnet as the armature revolves. The ohm. The pressure (electromotive force) produces a flow of electricity which meets with resistance in the conductor. field

Just as the frictional resistance in a water pipe opposes the flow of water, so the electrical resistance of a conductor opposes

The unit of resistance is the ohm (from mathematician who lived from 1789 to 1854). German Ohm, a the flow of electricity.

APPLIED MATHEMATICS

214

The ohm wire

is

nearly equal to the resistance of 1000

in diameter.

.1 in.

grees of resistance.

ft.

of copper

Different substances have different de-

The

resistance of metals increases slightly

as the temperature rises, but the resistance of carbon (incandes-

cent

lamp filament) decreases with a

rise in temperature.

Thus

the resistance of a 16 candle power incandescent-lamp carbon filament

is

about 220 ohms

400 ohms when

when

hot, but

it

may

be as high

as

cold.

Eesistance varies directly as the length and inversely as the cross section of a conductor.

of wire is

6

ohms

;

if

the resistance of 100

in.) is

of the

.3 in. in

ft.

same wire

diameter (cross

.6 in.

in diameter (cross section,

2 ohms.

The ampere. The unit lived

for measuring the rate of the electri-

An

ampere (from Ampere, a French from 1775 to 1836) may be defined as the

cal flow is the ampere.

who

ft.

ohms, the resistance of a wire of the

same material and length

physicist

if

the resistance of a wire

section, 7.07 sq. in.) is 8

28.27 sq.

Thus

2 ohms, the resistance of 300

is

current which an electromotive force of 1 volt will send through

a conductor whose resistance

The number

is

1 ohm.

of amperes of current corresponds quite closely

to the rate of flow of a stream of water.

•

We may

certain point in an electrical circuit the rate of flow

just as

we would say that

say that at a is

5 amperes,

at a certain point in a water pipe the

is 10 gal. per minute. Given an electromotive force of 1 volt, a circuit of 1 ohm resistance, and we have a current of 1 ampere. 115. Ohm's law. A very simple relation exists between the

rate of flow of the water

electromotive force, resistance, and current in a closed

V = the number of volts of electromotive R = the number of ohms of resistance, A = the number of amperes of current, V and we have Ohm's law, — = A. R Let

force,

circuit.

ELECTRICITY In words this law

may

215

The number of by the number of ohms of resistance gives the number of amperes of current flowing through a circuit. This law was first formulated by Ohm be stated as follows

:

volts of electromotive force divided

in 1827.

PROBLEMS 1.

How many

resistance if the

amperes are there in a circuit of 20 ohma generates 110 volts ?

dynamo

KK amperes. —V = —110 — = 5.5 F

C Solution.

2.

R

20

A battery sends

a current of 5 amperes through a circuit.

If the electromotive force •

is

10

volts, find the total resistance

of the circuit. 3.

If a cable has a resistance of .004

20 amperes es through 4.

it,

what

is

ohm and

a current of

the electromotive force ?

Find the resistance of an incandescent lamp which takes .5 ampere when connected to a 110-volt main.

a current of 5.

If a telegraph wire has a resistance of 200 ohms,

many amperes 6.

will be sent

The wires

in

an

through

110 7.

by a pressure

how

of 10 volts ?

stand 8 amperes

electric heater will

without becoming unduly heated. for

it

What must

be the resistance

volts ?

A dynamo generates

ance of the circuit

if

there

110 is

volts.

What

is

the total resist-

a current of 40 amperes ?

A

32 candle power lamp for a 220-volt circuit has a 330 ohms, and a 16 candle power lamp for a 110-volt circuit has a resistance of 180 ohms. Which lamp has 8.

resistance of

the greater current ? 9.

Construct a curve to show the relation between the

electromotive force and the resistance of a circuit in which

the current

10 ohms.

is

20 amperes, as the resistance varies from 1 to

APPLIED MATHEMATICS

216

If the electromotive force of a

10.

at

120

construct a curve to

volts,

dynamo remains constantly show the changes

in the

current as the resistance increases from 10 to 120 ohms.

In the preceding problems

116. Resistances in combination.

the resistance of the circuit ance, but in practical

was considered as a single

work the

made up

circuit is

resist-

of several

Thus in an electric lighting system the total resistance made up of the resistances of the dynamo, lamps, and connecting wires. The parts of a circuit may be combined in two

parts. is

distinct ways.

117. Series circuits.

When

the different parts of a circuit

are ed end to end and the whole current flows through each part, the circuit is called a series circuit.

Let

D

in Fig. 83 be a

dynamo

maintaining an electromotive force of

~7

110 volts measured across the termi-

/p\

/J\

AB. This means that 110 volts ^ -pm. 83 are continuously generated and used up in forcing the current through the circuit BCEFA Hence we may say that from B to A there is a drop in voltage of 110 volts. Let C, E, and F be arc lights having resistances of 4.2, 4.6, and 4.8 ohms respectively, and let the resistance of the line be 4 ohms. nals

.

= 4.2 + 4.6 + 4.8 + 4 = 17.6 ohms. — = -r=-z = 6.25 amperes.

Total resistance

By Ohm's At any point

law,

in the circuit the current

in a series circuit the current

is

is

constant.

6.25 amperes, since

But there

is

a con-

tinual drop in the voltage along the circuit as the voltage

used up in forcing the current along voltage, or drop of potential, as

Ohm's

law.

it is

is

path.

This drop in

sometimes

called, follows

its

ELECTRICITY The The The The

C = A R = 6.25 E = A R = 6.25 drop in lamp F = A R = 6.25 drop in the line = A R = 6.25 drop in lamp drop in lamp

Total

The arc lights in general use to

217

= 26.2 volts. = 28.8 volts. = 30.0 volts. = 25.0 volts. drop = 110.0. volts.

x 4.2 x 4.6 x 4.8 x 4

Chech.

light city streets are connected

and the entire current goes through each lamp. 118. Ammeter. The number of amperes of current is measured by an ammeter. It consists of a coil of wire suspended in series,

Fig. 84

between the poles of a magnet so that

it

rotates

through a

when the current es through. The coil carries light needle. a The instrument is graduated by ing through it currents of known strength, and marking on the scale the position of the needle. The type of ammeter commonly used is cut into the circuit when a measurement is made. small angle

119. Voltmeter. tial) is

Voltage (electromotive force, drop.of poten-

measured by the voltmeter. Most voltmeters are simply

special forms of ammeters.

by experiment.

It is

The voltmeter

put on circuits of

also

known

is

graduated

voltage and the

APPLIED MATHEMATICS

218

position of the needle

voltmeter

its

is

marked on the

scale.

In using the

terminals are connected to the ends of the parts

of the circuit in

which the voltage

is

to be

measured; the

Fig. 86

reading of the voltmeter

is

the

number

of volts of electro-

motive force, or drop of potential. If a voltmeter across the terminals of an arc light

is

and the reading

connected is

47

volts,

means that 47 volts are used up in running that arc light. In Fig. 86 the ammeter A is arranged to measure the current produced by the dynamo D and the voltmeter V is connected it

;

Fig. 86

Fig. 87

show ihe electromotive force between the terminals of the dynamo. Fig. 87 shows an ammeter and a voltmeter arranged to measure the current and drop in voltage in an arc lamp L.

to

ELECTRICITY

219

PROBLEMS Three wires having resistances of 2, 5, and 8 ohms respectively are ed end to end and a voltage of 90 volts is applied. How many amperes of current are there ? 1.

2.

Two

wires of resistances 6 and 8

ed in series. 3.

Two

If the current

is

ohms

respectively are

1.8 amperes, find the voltage.

incandescent lamps are in series and one has twice If the voltage is 110 and the an ampere, find the resistance of each lamp.

as great resistance as the other.

current

^ of

is

R = the resistance of one lamp. 2R = the resistance of the other lamp. V _ 110 _ 1

Solution. Let

R~3R~ 3

'

R = 110 ohms. 2R = 220 ohms. =

Total C/ieC

4.

R

330 ohms. 330

3

Find the internal resistance of a battery which gives

a

current of 1.5 amperes with an electromotive force of 5 volts, if

the external resistance 5.

An

is

1.33 ohms.

iron wire and a copper wire are in series.

If the

amperes, and the copper wire has a resistance of 3 ohms, find the resistance of the iron wire. voltage

6.

A

12

is

volts, the current 2.8

circuit consists

of

two wires

in series.

An

electro-

motive force of 30 volts gives a current of 3.2 amperes. If the length of one wire is doubled and the other is made 5 times as long, the current is .84 ampere. Find the resistance of the

two wires. 7.

What

amperes,

if

voltage

is

the circuit

necessary to furnish a current of 9.6 is

made up

of 2 mi. of No. 6

Brown &

Sharpe gauge copper wire (resistance of 1000 ft., .395 ohm) and 10 arc lights in series, each of 4.8 ohms resistance ? Find also the drop in voltage in the wire and in the lamps.

'

APPLIED MATHEMATICS

220 8.

A

dry

cell is

the wire in the bell cell 1

used to ring a door bell. The resistance of is 1.5 ohms, of the line .5 ohm, and of the

ohm. If the electromotive force of the

what current

will flow

when

the circuit

cell is 1.4 volts,

closed ?

is

9. What is the resistance per mile of No. 20 Brown & Sharpe gauge copper wire, if the voltmeter connected to the ends of 100 ft. of the wire reads 5.13 volts and the ammeter reads 5 amperes ?

10.

An

dynamo

arc-light

of 30

ohms

resistance supplies a

Brown & Sharpe gauge copper wire to a series of 50 arc lights, each adjusted to 47 volts. Find the electromotive force of the dynamo.

current of 6.8 amperes through 12 mi. of No. 6

The drop

Suggestion.

in voltage in the lamps

the drop in voltage in the total voltage is the

it

= 47 x 50. Find V = R-A. The

in the line by

of the drop in voltage in the three parts of

Check by finding the

the circuit. dividing

sum

dynamo and

total resistance of the circuit

into the total electromotive force

;

and

this should give 6.8

amperes.

In an

system there are 6 mi. of No. 6 wire, and 80 arc lights, each having a resistance of 4.5 ohms. The resistance of the dynamo is 3 ohms and the electromotive force is 3725 volts. Find 11.

electric lighting

Brown & Sharpe gauge copper

(a) the total resistance

in the

is

line,

;

(b)

the current

;

(c)

the fall of voltage

and lamps.

12.

The

110

If a voltmeter is connected across the mains in with a resistance of 6000 ohms, it reads 70 volts. What

series is

dynamo,

voltage across the mains of an electric-light circuit

volts.

the resistance of the voltmeter

?

Solution. Since there is a drop of 70 volts is a drop of 110 — 70 = 40 volts in

there

the resistance.

V= S

40

6000

1

=

150

ampere

-

in-

the voltmeter,

ELECTRICITY Since the current

is

the same in

221

all parts of

the circuit,

V= —

70

=

"'

10,500 ohms, resistance of the voltmeter.

150 Chech.

10,500

+ 6000 =

V

R=

16,500 ohms.

110

1

16700 = 150

ampem

13. A coil of wire is placed in series with a voltmeter having a resistance of 18,000 ohms across 110-volt mains. If the voltmeter reading is 60 volts, find the resistance of the coil of wire.

A

14.

voltmeter has a resistance of 10,000 ohms.

What

when connected

across

will be the reading of the voltmeter

110-volt mains with a of

5000 ohms

man having

a hand-to-hand resistance

?

120. Multiple circuits. When the branches of a circuit are connected so that only a part of the current flows through each of the several branches, the circuit called a multvple, parallel, circuit.

Fig. 89

is

or divided

shows three incandes-

cent lamps connected in multiple.

ordinary incandescent

The

£S p IG

.

89

lamps used in

houses are connected in multiple between mains from the terminals of the dynamo.

The

full electromotive force of the

dynamo, except the drop in voltage in the wires, acts upon each lamp but only a part of the current goes through each lamp. 121. To rind the total resistance of a multiple circuit. In Fig. 90 let the drop in voltage from B to C be 12 volts, and a and b have resistances of 2 and 4 ohms respectively. The pressure (electromotive force) in each branch is 12 volts just as in a water pipe of similar construction the pressure would be the same in each branch. „ ;

;

V R

12 "Tp

12

=

R 6 amperes in

a.

APPLIED MATHEMATICS

222

Hence the total current is 6 + 3 = 9 amperes. The total resistance of a and b is given by T,

iJ

= 4 0llmS = V = 12 -9" 1.

X

-

3

We will now obtain a general f ornrala for the total resistance of a multiple circuit.

Let

V—

the drop in voltage from

r = resistance of a. r = resistance of R = total resistance. V= current in a. x

b.

2

V=

current in

b.

B

to C.

ELECTRICITY

When

223

two, three or four equal resistances are combined in 2

T

7*

multiple,

we have from

(1), total resistance

= r— =

from


= ^— = ^

from


= -j— =

Thus when ten 16 candle power lamps

122. Graphical total resistance of

method

: :

>

-

2

>

-.

a

ohms

of 220

are connected in multiple the total resistance

220 io

-z

resistance

is

22 ohms.

of finding the total resistance.

The

a multiple circuit can be readily determined

by a graph. EXERCISES 1.

Construct a graph to find the result of combining

ances of 20 and 30

ohms

OX any convenient length, and with

Take

OM = 30 ohms, ohms. Draw secting at A.

is,

interON AB = 12 ohms, the

AB = OM-XN

OM+XN

Y

25

(1) SO

and XBA,X0M give two equations. Eliminating XB from

OXN;

these equations gives (1).

15

IO

14, p. 77.)

total

number 20, 30,

5

(See

A similar construction gives the

off

30

Prove geometrically. The two pairs of similar triangles OB A,

Problem

convenient units lay

XN = 20

and and XM,

total resistance.

That

resist-

in multiple.

resistance

of

p IG

92

any

of resistances in multiple.

Thus, given the resistances

and 18 ohms, combine 20 and 30 ohms,

as above.

Then

APPLIED MATHEMATICS

224 lay

XP =

off

18 ohms.

PB, intersecting

CD 2.

is

XA

Draw

at C,

30

the total resistance.

What

resistance

must be

combined with

24 ohms

to

obtain a total

resistance

of

8

Y

and

OX

IO

any

convenient

length and with a convenientunit lay off

EO 15

ohms? Take

25

OM = 24 ohms.

MX. On such that'

MX

Draw

a

o

take a point A,

AB =

8 ohms.

Draw

OA, and extend to meet XP at = 12 ohms, the required resistance. N. The graphical method should be used to solve and check some

XN

the following problems.

PROBLEMS 1. Three resistances of 20, 30, and 40 ohms are ed in multiple. Find


20-30-40

of

ELECTRICITY 4.

Two lamps

each other,

and 150 ohms

of 100

and the

pair

is

225 are put in parallel with

ed in series with a lamp of

100 ohms.. If the electromotive force

is

200

volts,

what

will

be the current? 5.

A

known

resistance of 10 resistance.

ohms

put in parallel with an un-

is

If an electromotive force of 120 volts gives

a current of 20 amperes, find the

Solution. Let

r

10 r

Then

+

10 120

unknown

= the unknown =

resistance. resistance.


r 6

= the

total resistance.

20

+

r

10

r

10

r

10 x 15 Check.

10

+

15

= = _ -

60

+

6

r.

15 ohms.

150 6

ohms.

25

6. A lamp of unknown resistance is put in parallel with a lamp of 220 ohms resistance. If a voltage of 110 volts gives a current of 1.6 amperes, what is the unknown resistance ? 7.

The

ohms.

total resistance of three wires in multiple is 1.52

If the resistance of two of the wires

respectively, 8.

The

4.8 ohms,

what

is

3 and 5

is

ohms

the resistance of the third ?

total resistance of two conductors in multiple is and the sum of the two resistances is 20 find them. ;

Pig. 95 9.

The

resistance between Find the resistance x.

total

5.25 ohms.

A and B

in

Fig. 95 is

APPLIED MATHEMATICS

226

Three resistances in parallel are in the ratio 1:2:3. an electromotive force of 120 volts gives a current of

10.

If

11 amperes, find each resistance. 11.

Twenty 16 candle power

If the resistance of each resistance, 12.

A

and what

is

lamp

110-volt lamps are in multiple. is

220 ohms, what

110-volt incandescent lighting

three multiple circuits of

;

(b)

the total

circuit divides

into

and 10 lamps respectively. If 220 ohms, find (a) the resistance

5, 8,

the resistance of each lamp of each branch

is

the current ?

is

the total resistance

;

(c)

the current; (d) the

current in each branch. 13.

Construct a curve to show the change in the resistance

of a multiple circuit consisting of a

number

of incandescent

lamps of 220 ohms, as the number of lamps increases from 1 to 20. 14.

Two conductors of 12 and 18 ohms respectively are in What resistance must be placed in series with the

multiple.

multiple circuit to give a current of 5 amperes with an electro-

motive force of 110 volts

?

WOKK AND POWER 123.

The watt. When an

electromotive force overcomes the

and causes a current to flow, work is done. This is analogous to the case where work is done by the pressure of steam on the piston of an engine. The number of pounds pressure multiplied by the number of feet through which the piston is moved gives the number of foot pounds of work. Power is the rate of doing work. The unit of mechanical power is the horse power, the rate of doing work equal to 33,000 ft. lb. per minute. The unit of electrical power is the resistance of a conductor

watt (James Watt, Scotland, 1736-1819, practically the inventor of the modern steam engine), the rate of doing work equal to 44£

ft. lb.

per minute.

;;

ELECTRICITY Power in watts equals the number of number of amperes.

227 volts multiplied

by the

W=V-A. Thus

a

if

dynamo

(1)

supplies a current of 50 amperes at a voltage

of 110 volts, the power delivered

From Ohm's law

—V = A,

110 x 50

is

may

(1)

= 5500 watts.

be written

W = A R. *=*" 2

In words

:

(2) (3)

watts equal volts multiplied by amperes

(1)

watts equal current squared multiplied by resistance

(2)

watts equal volts squared divided by resistance.

(3)

To express watts in horse power : Since

1 h. p.

124.

is

33,000

ft. lb.

per minute,

1 watt

—

-.

a watt

=

= 44J ft. lb. per minute, 33,000 ., 1 h. p. = —rwatts. 44i 1 h. p. = 746 watts.

and

The kilowatt. For many purposes a larger unit than convenient. Hence 1000 watts, called a kilowatt (kw.),

is

sometimes taken as the unit of power. 125.

The kilowatt hour.

used in measuring

by 1 kw.

in 1 hr.

5 kw. for 4

hr.,

Thus 20 kw.

1 kw. for 20

kilowatt hour It is the

is

a practical unit

energy expended

might mean 2 kw. and so on.

hr.

hr.,

1 u 1 kw. hr.

= 44£ x 1000 x 60 ft. lb. = 33,000 x 60 ft. lb. 44i x 1000 X 60 ^ = h. p. hr.

1 kw. hr.

=

1 kw. hr.

1 h. p. hr. tt Hence

A

electrical energy.

1

^

y 6Q

1.34 h. p. hr.

for

10

hr.,

APPLIED MATHEMATICS

228

PROBLEMS 1.

An arc

power does

W=

V-A =

= 6h -

746

-

p

45 x 10

=

450 watts.

-

A

16 candle power incandescent lamp is on a 110-volt and takes \ ampere. How many watts per candle

circuit

power are required Solution.

3.

40 kw. 4.

How much

volts.

absorb ?

Solution.

2.

10 amperes at 45

light requires

it

A

?

W=

V-A =

55 — =

3.5 watts per candle power.

dynamo has a

110 \ •

=

55 watts.

voltage of 550 volts and

How many amperes in the current ? How many watts will be lost in forcing

through the armature of a dynamo, and the current is 30 amperes ?

if

is

producing

the current

the resistance

is

.035

ohm

5. A 150-kw. dynamo was supplying 273 amperes. What was the voltage of the dynamo ? 6. How many horse power are required to send a current of 65 amperes through 10 mi. of No. 6 B. & S. gauge copper wire ? 7.

A

current of 15 amperes flows through 100

wire whose resistance lost in the 8.

9.

\

10.

per foot.

ft.

How many

of iron

watts are

a current of 50 amperes 450 watts are absorbed in

Find the drop

A voltmeter

in voltage in the conductor.

has a resistance of 17,000 ohms. If placed

a circuit of 110 volts,

operate

ohm

wire ?

With

the conductor.

in

is

how much power

is

required to

it ?

A 200-volt lamp takes \ ampere. How many watts are re-

quired for 30 such lamps ? to drive the

dynamo

if it

How many horse power are required has an efficiency of 90 per cent ?

ELECTRICITY Solution.

200 x £ x 30

=

—— = 68 —— = 2

229

2000 watts. 2.68 h. p. 100 per cent efficiency.

'

2.98 h. p. 90 per cent efficiency.

In a room there are thirty 16 candle power incandescent and 3 arc lights, each taking 6.8 amperes at 50 volts. How many watts and how many horse power are required to operate these lights ? 11.

lamps, each taking .52 ampere at 110 volts

12.

How many incandescent

of 220

lamps, each having a resistance

ohms and requiring a current of

by a 10-kw. generator

;

.5

ampere can be run

?

13. A 25 h. p. dynamo is running at 550 volts. How many amperes in the current ? How many 16 candle power incandescent lamps can be placed on the circuit if each lamp takes

55 watts and there 14.

In an

is

a loss of 10 per cent on the line

electric-lighting circuit there are

60 arc

?

lights,

each

taking 50 volts, and 15 mi. of wire having a resistance of 2.1

ohms per

mile.

If the current

is

9.6 amperes,

how many

watts are required to run the lights ? 15. Find the energy in foot pounds expended per candle power in a 16 candle power incandescent lamp in 1 hr., if it takes £ ampere at 110 volts. 16.

how many

pended in 1 hr. Problem 15 ? 17.

many 18.

power are light requires 50 volts with foot pounds per candle power are exHow does this compare with the result in

If a 500 candle

9.6 amperes,

?

A 10-kw. dynamo

has an efficiency of 88 per cent.

horse power are required to drive

The

efficiency of a

dynamo

is


16 candle power lamps on the ampere at 110 volts ?

85 per it

circuit,

How

it ?

cent.

when

How many

there are 200

each lamp taking £

APPLIED MATHEMATICS

230 19.

How many

amperes at 120 volts must be furnished a is to lift 900 lb. 70 ft. per minute, if it of 70 per cent ?

hoisting motor which

has an efficiency 20.

A

motor operates a pump which in 1

hr. lifts

20,000

gal.

=

of water (1 gal. 8^ lb.) 400 ft. If the combined efficiency of the pumping system is 72 per cent, what current will the motor

require at 550 volts ?

An

with its load weighs 8 T. on a 20 lb. per ton. How much power is necessary at the axle to move the car 10 mi. per hour ? If the motor and gearing have an efficiency of 75 per cent, how many amperes are required on a 550-volt circuit ? 21.

electric street car

;

level track the pull required is

To perform a certain amount of work 30 kw. hr. are If the dynamo gives a current of 125 amperes at 220 volts, how long must it be used to perform this work? 22.

required. '

— —— = „_..kw.

125x220 —

_

Solution.

-^- =

27.5

1.09 hr.

27.5

23.

What

A

5-kw. motor

is

used to operate a printing press 8 hr. power at 12 cents per kilowatt

will be the cost of the

hour? 24.

What

15 amperes 25.

power

An

is

at

the cost of running a motor which requires volts, at 12 cents per kilowatt hour ?

110

incandescent lamp takes

.6

ampere at 110 volts. what is the cost

If

on a current of lamp 12

6.6

costs 15 cents per kilowatt hour,

of

operating the lamp 12 hr. ? 26.

An

amperes. at

inclosed arc

lamp takes 80

How much

does

15 cents per kilowatt hour

How many

volts

cost to operate the

it

hr.

?

watts per candle power are required in each of the following lamps ? If power costs 10 cents per kilowatt hour, how much would it cost per hour to keep each lamp at 27.

ELECTRICITY

231

candle power ? Construct a curve to show the relation between the candle power of each lamp and the cost per

full

candle power.

Volts

APPLIED MATHEMATICS

232

PROBLEMS 1.

Find the heat

due to a current of 60 amperes through

loss

a resistance of 10 ohms.

W=A

Solution. 2.

A

R=

2

=

60 2 x 10

36,000 watts in 1 min.

conductor having a resistance of 5 ohms carries a curHow much heat is developed in 1 hr. ?

rent of 18 amperes. 3.

How much

(b) in calories ? 4.

A

5.

developed in a wire of 15.2 ohms

is

(c)

What

is

is

sent over a line of 2 ohms

the drop in voltage?

hour (a) in watts

A

re-

min., (a) in watts ?

in British thermal units ?

current of 36 amperes

resistance. loss per

heat

by a current of 8 amperes in 15

sistance

?

(b) in British

What

is

the heat

thermal units ?

current of 12 amperes flows through a resistance of

ohms

for 15 min., and another current of 8 amperes flows through a conductor of 2.5 ohms resistance. How long must the second current flow in order that the amount of heat gen-

3.2

erated 6.

may

be the same as in the

first

case ?

Construct a curve to show the heat loss in a conductor from 1 to 10 ohms while the current

as the resistance changes

remains constantly 5 amperes. 7.

of 1

Construct a curve to show the heat loss in a conductor as the current varies from 10 to 20 amperes.

ohm resistance

8.

In a conductor of 10 ohms resistance the voltage increases volts. Construct a curve to show the heat loss.

from 10 to 1000

A

Leelanche cell used to ring a doorbell has an electro9. motive force of 1.6 volts and the current is .75 ampere. If the wire has a resistance of .4 ohm, what per cent of the power is

the heat loss in the line ?

Solution.

1.6

A

1

R

x

=. .75 2

.75

x

.4

= =

—= •

23

1.2 watts, total power. .23 watts,

heat loss in

in per cent. 19

line.

ELECTRICITY 10.

233

The dynamo

of an arc-light system furnishes a current 3000 volts. The circuit is made up of 16 mi. gauge copper wire. What per cent of the power

of 9.6 amperes at

& S.

of No. 6 B. is

the heat loss in the line ? 11.

What

A is

6-in.-plate stove requires 5.5

the cost of running

it

amperes at 110 volts. 30 min. if the current costs

6 cents per kilowatt hour ? 12.

Find the

cost of heating a 6j-lb. flatiron for 3 hr., if

it

takes 4 amperes at 110 volts, at 6 cents per kilowatt hour. 13.

An

radiator takes

electric

Find the cost for 8

13.6 amperes at 110 volts.

hr. at 6 cents per kilowatt hour.

In an electric heater there is a coil of iron wire 224 ft. having a resistance of \ ohm per foot. If it is connected to a 110-volt circuit, how much heat is generated ? 14.

in length

15.

make an electric soldering iron to be German silver wire of resistance How many feet will be required to give

It is desired to

heated by a coil of No. 27 1.25

ohms per

foot.

200 watts on a 500-volt

circuit ?

Wiring for Light and Power The

mil.

In

eter of wire, the

mil

127.

electrical calculations involving the is

1 mil

A

diam-

usually taken as the unit of length.

=

circular mil is a circle

.001

in.

whose diameter

is

1 mil.

A circular mil = tr r = 7r x .5 = .7854 sq. mils. 1 circular mil = .7854 sq. mils. 1 sq. mil = 1.273 circular mils. 2

2

Circles are to each other as the squares of their diameters.

Hence

to find the area of a circle in circular mils, square diameter expressed in mils.

A

mil foot of wire

is

1

ft.

long and 1 mil in diameter.

its

In

practice the resistance of 1 mil-ft. of copper wire is usually

taken as 10.7 ohms.

APPLIED MATHEMATICS

234

PROBLEMS

The diameter

1.

mils

;

How many

2.

of a wire

Find

is \ in.

(a) its diameter in

(b) its cross section in circular mils.

diameter (a) \

circular mils in the cross section of a wire of

.125

in. ? (b)

.06 in. ?

in. ? (c)

Find the diameter and area in square mils of a wire whose cross section is (a) 10,381 circular mils; (b) 26,250 3.

circular mils

A

4.

;

105,590 circular mils.

(c)

copper bar

1

is

section in square mils

in.

and

by \ in. Find the area of a cross in circular mils.

Find the resistance of 1000

5.

ft.

of copper wire 40 mils in

diameter.

Solution.

The

cross section

Resistance of Resistance of 1

1 ft.

mil foot

= 40 2 = 1600 circular = 10.7 ohms.

of wire of 40 mils diameter

10 7 00669 ohm 1600 of wire of 40 mils diameter = .00669 x 1000 = 6.69 ohms.

= £p- =

Resistance of 1000

ft.

Find the resistance of 1000

6.

diameter of (a) 460 mils mils 7.

;

(e)

A

mils.

(b)

;

ft.

-

-

of copper wire that has a

289.3 mils

;

(c)

.1 in.

;

(d) 40.3

20 mils.

current of 75 amperes

wire 229.4 mils in diameter.

Solution. Cross section Resistance of 1

ft.

=

is sent through 1 mi. of copper Find the drop in voltage.

229.4 2

=


= —10—7— = .000203 ohm. 52,620

Resistance of 1 mile

V= A R = 8.

What

is

75 x 1.07

= .000203 x 5280 = 1.07 ohms. = 80.3 volts.

the drop in voltage in a circuit of 5 mi. of copper

wire 162 mils in diameter

if

the current

is

40 amperes

?

— ELECTRICITY With a

9.

235

current of 210 amperes -what will be the drop in ft. of copper wire 460 mils in diameter ?

voltage in 2500

How many

circular mils are required in a power line long with a current of 150 amperes, if a drop of 12 volts allowed ? 10.

500 is

ft.

Solution. Let n 10.7 x 500 5350 "

= number of = 5350 ohms

per circular mil for 500

= number

ohms

for

5350 x 150

of

500

,

= number

circular mils required. ft.

resistance of n circular mils

ft.

,,,..,

of volts in drop.

n

From which

12.

=


=

.00016 ohm, resistance of 1

= =

.08

ohm, resistance

150 X .08

the above solution

is

=

of 500

ft.

of the wire.

ft.

12.

we may

obtain the following formula,

in general use for finding the size of conductor required

to carry a

given load.

,T JNo. circular .

=

..

.,

mils

= 21.4

x

distance one r-

way /

in feet

x amperes

;

volts lost 11.

A motor is 300

ft.

from the dynamo.

How many circular

mils are required for a current of 90 amperes,

6 volts

is

if

a drop of

allowed ?

Find the number of circular mils required to deliver

12.

10 kw. to a motor at a distance of 200 ft., with 100 volts pressure at the motor, if a drop of 5 volts is allowed in the line. 13. Find the number of circular mils required to transmit 25 kw., with a 20 per cent drop in the voltage, a distance of 10 mi., if the voltage at the load is to be (a) 100 volts (b) 500 volts (c) 1000 volts. ;

;

Suggestion.

—— = 125 volts .80

at the

dynamo.

APPLIED MATHEMATICS

236

14. It is required to deliver 120 h. p. to a motor 2 mi. away, from a dynamo which has a voltage of 550 volts. If the line loss is to be not more than 16 per cent, find the cross section of the wire in circular mils and the number of pounds of

copper required.

Dynamos and Motors 128. Construction.

When

a closed wire

is

rotated between

the poles of a magnet so as to cut the lines of force, a current flows in the wire.

The dynamo is The

constructed on this principle.

armature

is

the part of the ma-

chine in which the current erated,

and

in

is

gen-

/'

[

\ \ \ \\ \

\.

most machines the

armature revolves.

The

field

is

the space between the poles of the

magnets in which the armature revolves. The magnets are pieces of soft iron, which are magnetized by a current from the machine itself or from a separate dynamo. This current flows in coil's which are placed around the magnets. In Fig. 96 the armature is represented

by a

single wire revolving in the

field.

The field coils connected in three ways. dynamos generally excite their own fields and ways of connecting the field-magnet coils. 129.

;

1.

The field-magnet

Series-toound dynamos.

coils are

Direct current there are three

connected to the armature so that

the whole current generated es through Fig. 97

them. 2.

Shunt-wound dynamos. The field-mag-

net coils are connected in multiple with the terminals of the armature

;

part of the current goes

These

coils consist of

many

hence only a

through them.

FlG> 9°

turns of comparatively fine wire.

ELECTRICITY

237

Compound-wound dynamos. The field magnets are wound with two sets of coils, one in series and 3.

one in multiple with the armature. Motors are also wound in these three

way g

'

Fia. 99

130. Electrical efficiency of

and motors. Since

it

dynamos

requires pressure (voltage) to drive a

and field coils, there is a loss power in a dynamo and in a motor. This loss is sometimes

current through the armature of

called the copper loss.

only the copper .

,

_,

Electrical efficiency takes into

loss. „

.

Electrical efficiency J of a .

of a Electrical efficiency J

dynamo J

Power =—

motor

=

given out

——r

Power generated Power left for useful work -=r .. Power supplied to motor ,

PROBLEMS 1. The output of a series-wound dynamo is 5 kw. at a voltage of 110 volts. The resistance of the armature is .06 ohm and of the field coil .072 ohm. Eind (a) the copper loss (b) the elec;

trical efficiency

—=

;

(c)

=

the total electromotive force generated.

45.5 amperes.

.072 = .132 ohm, total resistance, A 2R = 45.5 2 x .132 = 273 watts, copper loss. Fig. 5000 + 273 = 5273, total power generated. (P) \ f fir = 95 per cent, electrical efficiency. (c) V=AR = 45.5 x .132 = 6 volts, loss in armature and field coils. 110 + 6 = 116, total electromotive force generated. .06

+

(a)

2.

A

100'

series-wound motor has a resistance of .68 ohm.

supplied with 15 amperes

at

a

voltage

When

of 105 volts,

find

APPLIED MATHEMATICS

238

(a) the copper loss

;

(b)

the electrical efficiency

(c)

;

the volts

lost in the motor. Suggestion. Find the copper loss as in Problem 1 and subtract from the number of watts supplied to the motor. Electrical efficiency = ^fff Drop in voltage = 15 x .68.

it

.

A

shunt-wound dynamo furnishes 5 kw. at a voltage of The shunt resistance is 45 ohms and the armature resistance is .06 ohm. Find (a) the copper loss (b) the electrical 3.

110

volts.

;

efficiency.

A =

Solution.

W = 45.5 amperes. —

SOOOW/ITTC i/oyaLr<5

In the shunt,

V= A=— R .

45.5

+

110 2.44

W = VA = 110 x 2.44 W = A*R = 47.9 x a

=

45

.06

(6)

+

405

4Ito

2.44 amperes. l

Fig. 101

= 47.9 amperes. = 268 watts, loss in shunt. = 137 watts, loss in armature.

(a)

500Q

...

= =

405 watts, total loss. 5405 total watts. 93 P er cent, electrical

efficiency.

4. The armature of a shunt motor has a resistance of .02 ohm, and the shunt a resistance of 62 ohms. If the input is 5 h. p. at 124 volts, find (a) the copper loss (b) the electrical efficiency. ;

Solution.

ELECTRICITY 5.

A

50-kw., 125-volt,

239

compound-wound dynamo has a shunt ohm, and

resistance of 62.5 ohms, a series-coil resistance of .001

armature resistance of .002 ohm.

•an

Compute the copper

losses

electrical efficiency.

A

Solution.

?££

a

=

>

J^ /2ffKX73

* g ^~f^

400 amperes.

— = —62.5 — = 2 amperes.

In shunt,

soooo watts

^^

%$joog ^§ >^v

and the

<

FlG 103 -

R,

400 + 2 402 2 x .002 402 2 x .001 125 x 2

= 402 amperes, total current generated. = 323 watts, loss in armature. = 162 watts, loss in series coil. = 250 watts, loss in shunt. 735 watts, total

a

Note that the the

sum

8 "

a §

=

loss.

98.6 per cent, electrical efficiency.

total current

generated by a shunt dynamo equals and in the field coils.

of the currents in the armature

6. A compound motor is supplied with 50 amperes of current from 110-volt mains. If the armature resistance is .09 ohm, the

series-coil resistance .078

ohm, and the shunt-coil resistance

55 ohms, find (a) the copper loss Solution.

50 x 110

=

110 —v = — —= it

Find

55 50 — 2

=

;

(b)

the electrical efficiency.

5500 watts.' o 2 amperes

m shunt. •

i.

*

48 amperes in armature.

armature, and series coil to be 220, 207, and 180 watts respectively, and the electrical efficiency 89 per cent. 7.

The

loss in shunt,

The output

of a series

dynamo is 20 amperes at 1000 volts. is 1.4 ohms and of the field coil

resistance of the armature

1.7 ohms.

Find the copper

loss,

the electrical efficiency, and

the volts lost in the dynamo. 8. The armature of a shunt motor has a resistance of .3 ohm, and the shunt a resistance of 120 ohms. When running at full load on a 110-volt circuit the motor takes a current of 8 amperes. Find the copper loss and the electrical efficiency.

:

240

APPLIED MATHEMATICS

Find the copper

losses

and

electrical efficiency of the follow-

ing dynamo-electric machines

Dynamos

No.

ELECTRICITY

241

PROBLEMS

A

1.

motor

supplied with a current of 20 amperes at

is

110 volts. If 2.8 h. p. are developed at the pulley, find the commercial efficiency of the motor. Input

Solution.

Output

Commercial

efficiency *

110 x 20 X

Check.

.95

= =

110 x 20 watts. 746 x 2.8 watts.

—

=

110 x 20 95 per cent.

= = 2090 watts =

2.8 h. p.

A

2.

volts

motor generator takes a current of 14 amperes at 220 and supplies a current of 112 amperes at 25 volts. Find

its efficiency. 3.

270

A 220-volt ft.

electric hoist is raising coal at the rate of 1 T.

per minute.

If the current

is

90 amperes, what

is

the

efficiency of the hoist ? 4.

A

ciency 5.

3-kw. motor if it

The output

to drive 6.

A

it,

is

used to operate a

lathe.

Find

its

effi-

takes 30 amperes at 110 volts.

what

of a generator is its

is

50 kw. If

it

requires 76 h. p.

efficiency ?

550-volt generator supplies a current of 300 amperes.

If the generator has

an efficiency of 85 per cent,


it

how many

?

It takes 25 h. p. to operate a dynamo which supplies power 40 arc lights in series at 7 amperes. The resistance of each lamp is 8 ohms and the line resistance is 25 ohms. Find the efficiency of the dynamo. 7.

for

8.

A

lighting circuit consists of 1200

ft.

of No.

6B.&S.

gauge copper wire and eighty 16 candle power incandescent lamps in multiple, each having a resistance of 220 ohms. If the voltage

is

110 at the lamps and 7.5

the generator, find

its efficiency.

h. p. is

supplied to

APPLIED MATHEMATICS

242 9.

In testing a motor the following results were obtained. efficiency given by each test.

Find the No.

CHAPTER

XVIII

LOGARITHMIC PAPER 132. Description of logarithmic paper. In

problems where a formula,

it is

many engineering

necessary to compute a set of values from

found that the required values can be secured

it is

quickly and easily by using paper ruled on the logarithmic scale. This paper is used both as a " ready reckoner," to read of values

off tables

and

to find the

lies

in the fact that

law connecting the two

The advantage

variables in the problem.

many

of logarithmic paper

formulas which are represented by

curves on squared paper are represented by straight lines on

Hence while many pairs of values must be worked out to construct a curve on the former, only two or logarithmic paper.

three pairs are required for the latter.

104 shows the way in which logarithmic paper is ruled. aj-axis and the y-axis are laid off in divisions exactly like those of the slide rule. That is, OX and OY are each divided Fig.

The

into

1000 equal parts

(log

2

0.477)

and so

= ;

0.301)

4

is

;

3

is

;

2

is

placed at the 301st division

= = 0.602),

placed at the 477th division (log 3

placed at the

602d

division

(log

4

on.

Exercise.

Construct a graph to read off the area of a circle of

any given radius. In order' to learn the properties of logarithmic paper we will

by locating points. Later it will be shown graph can be constructed easily by locating only

construct the graph that the whole

one point. 243

:

APPLIED MATHEMATICS

244

The formula

Radius

Area

for the area,

1

1.2

3.14 4.52

.

1.6

7.07

a

—

2

2

77-r

,

gives the following table

4

3

12.6 28.3 60.3

5

6

7

78.5

113

154

10

201

8

314

S

lOjj

Fig. 104'

Locating the points as shown in Fig. 104, we see that the points

lie

on the straight lines AB, CD, and EF. Hence From it we see that is the graph required.

AB — CD — EF when

the radius

the radius

is 3.5,

is

2.5 the area

and so

on.

is

19.6

;

when

the area

is 38.5,

LOGARITHMIC PAPER

245

Some properties of The equation a = irr 2 is in the

133. Properties of logarithmic paper. the paper

form y

may now

= mxn

-

be noted.

AB, CD, and

EF

are parallel to one another.

FX = — the exponent of BD=CE = \YZ FX = 2 EX; hence — EX 2,

r.

The graph can be drawn mechanically as follows Find P, Tack the sheet of paper on a drawing and P. Set board so that the T-square, in position, lies on the T-square on A (making OA = 3.14) and draw AB. Set the T-square at C on OX directly below B and draw CD. Similarly, :

the mid-point of YZ.

draw EF. Check;

F

should be directly opposite A, that

is,

FX = 3.14. It will be

found that these are general properties of logarithmay be used to construct graphs for formulas

mic paper, which of the

form y

= mxn

;

that

is,

a formula in which y equals an

expression consisting of only one term in which the variable is

raised to

any power

(n,

being positive, negative, or fractional)

and multiplied by any number.

This form alone will be con-

and some of the properties which lead to simple and accurate constructions

sidered in the following discussion, of the paper

will be considered.

I.

Equations of the Fokm y

= mx

246

APPLIED MATHEMATICS

It is seen that all these lines are parallel.

When we

plot

y = x (1) and y = 2x (2), we are really plotting the logarithmic equations log y = log x (V) and log y = log 2x, or log y = log x + log 2 (2'). Comparing (1') and (2'), we see that they

Fig. 105

by the constant term log 2 on the right side that every point of the graph of (2) is 2 above the corresponding point of the graph of (1). Note that the graph of each of differ only

;

is,

=

and all x, is made up of two lines these equations, except y the lines are parallel to OZ. Hence to graph any equation of the form y

;

= mx, for example, y = 5x, proceed as follows.

Prom

LOGARITHMIC PAPER 5 on

PQ

247

OF draw MN parallel to OZ. Take OP = YN and draw P to 5 on XZ. MN—PQ is the required graph.

from

We

graph we are two or more parallel lines, and that one line in each graph cuts OX and XZ or OX and OY. Thus in the graph of y = 5x, PQ, cuts OX and

The

slope of a graph.

to consider (except

XZ.

We will

y

—x

XQ

call

^-^ the

Xr

shall find that each

and y

= or ) 1

slope of the

consists of

graph ; that

is,

the tan-

gent of the angle which the line makes with OX, always taking the angle on the right-hand side of the line.

II.

A.

When n

Equations of the Form y is

a positive whole number.

= mx"

248

APPLIED MATHEMATICS

from 2 on OY to OA, 1.41 for the square root of 20 read from 2 on OF to BZ, 4.47. Similarly, y = a; 8 gives cube roots OD gives the cube root of numbers containing 1, 4, 7 figures, FG ;

;

•

lO

•

•

;

LOGARITHMIC PAPER

249

Note that each y is twice as great as the corresponding y in 2 On locating the points and drawing the lines of the y=x graph it. will be seen that the lines are parallel to the lines of 2 2 y = x and 2 units above them. Hence the graph of y = 2 x .

be constructed mechanically as follows Tack the sheet of paper on a drawing board so that the edge of the T-square, in

may

:

and the mid-point of YZ. Move the T-square and draw a line from 2 to YZ. Move the T-square to a point on OX directly below the point already determined on YZ and draw a line to YZ. Continue in the same manner and the graph will end at 2 on XZ if accurately drawn. This method holds for all cases where x" has a coefficient. Note that the exponent of x is the slope of the graph.

position, lies

up

to 2

5.

on

on

OY

Construct

y

= 1.68 x

B.

When n

(c)

2 ;

is

graph

the (d)

y

of

= .0625 x

(a)

3 .

a positive fraction.

y

= 2 x*

;

(b)

y

= .5 x*

APPLIED MATHEMATICS

250

into 5 (the denominator of the exponent) equal parts,

make the

the points so as to

slope

and

§.

= 5 xi, start the graph at OA, thus making the slope |.

If x has a coefficient, for example, y

5 on

OF and

draw

it parallel. to

Fig. 107

2.

3z§; 3.

(d)

y

= 5x

1 -i ;

(e)

y

y = 2 xi = .06x™.

(b)

y

;

(c)

y

=

Construct a graph to show the distance ed over by a

falling 4.

= x\\ = 2.5x™; (/)

Construct the graphs of (a) y

body in 1 to 10

sec.

Construct graphs to find (a) the surface,

of spheres of radii

from 1 to 10

in.

(b)

the volume

LOGARITHMIC PAPER When n

C.

is

negative.

EXERCISES 1.

Construct the graph of y

X

= x _1

or

y

=

.251

:

:

APPLIED MATHEMATICS

252 begin to draw

it

from m on OY. Thus,

graph y

to

=

4 ar 1 from ,

OF draw a line parallel to YX cutting OX at a point A B on YZ directly above A draw a line parallel to YX

4 on

;

from

cutting

XZ

at C.

=

arf. Construct the graph of y Divide OX and YZ into 2 (numerator of the exponent) equal 2.

and

parts,

OY

graph 3.

YA

and

Draw

equal parts.

XZ

into 3 (denominator of the exponent)

lines as

shown in Kg.

108, and

we

get the

-BC-DE-FX.

Construct the graphs of

= 28x-\ = 125 x2i (g)y = .006x(h) y = 2800 ar

= 3x~\ = .5xy = 4:X-i. y=Sx-

(e) y

(a) y

2

(b) y (c)

(d)

3- 5

(/) y

.

-

.

1 - 12

i.

.

PROBLEMS 1. If in a gas engine the gas expands without gain or loss 1 23 = 3060. Conof heat, the law of expansion is found to be pv struct the curve to show the pressure as the volume increases -

from 10

cu. in. to 26 cu. in. Locate only one point (Fig. 109) when v = 10, p = 180. Mark this point by A on OY. The exponent of v is — |ff, when ;

the equation

is

in the form

-1 p = 3060 « 28 -

.

on OY, and ON = 100 mm. on OX. Measure Tack the paper on a drawing board so that the T-square, in and N. Move the T-square to A and draw position, lies on AB. Move the T-square to C on YZ directly above B and draw CD. AB — CD is the graph from this graph pressures can be read off for volumes from 10 cu. in. to 100 cu. in. Given that steam expands without gain or loss of heat; construct graphs on logarithmic paper for volumes from 10 to 100 cu. in.

OM = 123 mm.

M

;

2. 3.

pv 1 11 pv 1M -

= 3000. = 2840.

4.

pvtt

5.

pv

1 - B1

= 3200. = 3420.

6.

pv 9 = 250.

LOGARITHMIC PAPER

253

The diameter d of wrought-iron shafting to transmit h power at 100 r. p. m. is given by d = .85 h\. Construct graph and make a table for horse power from 10 to 80.

7.

horse the

,23 ftv'

8.

The number of

= 30SO

gallons of water per minute flowing over

a rectangular weir 6 in. wide

= the

number

is

=

given by g

M, where

17.8

= the

of gallons per minute, and h

depth in from the level of free water to the sill of the weir. Construct the graph and make a table showing the number of 6 in. gallons per minute for depths 1, 1.5, 2, 2.5, g

inches

•

•

,

APPLIED MATHEMATICS

254 9.

of cubic feet of water per minute discharged

The number

is given by Q = 18.5 bh\, where Q = the number of cubic feet per minute, b = breadth of notch in feet at the free surface, and h = depth in inches from the free level to the bottom of the notch. Construct a graph and make a table for the quantity of water discharged for depths from 6 to 15 in. when b = 1 ft.

over a V-notch, or triangular weir,

10.

The diameter

of a copper wire

given by d

electric current is

and A

eter in inches,

= the

which

will be fused

by an

= .00212 Ai, where d = the diamnumber

of amperes.

Construct a

graph and make a table of diameters of wire which will be fused by currents of 10, 20, 30, 100 amperes. •

11.

The weight

•

•

,

in pounds that a rectangular steel beam,

ed at both ends, can sustain at its center is given by bd 2 w = 890 where w = the weight in pounds, b = the breadth

—

)

d = the depth Of beam in inches, and I = beam in feet. Find the number of pounds that can be ed at middle of a steel beam 4 in. in breadth and 15 ft. long of

beam

in inches,

the

length of

depths from 4 to 10 12.

the for

in.

In accordance with the building laws of Chicago the safe

load in tons, uniformly distributed, for yellow-pine beams

given by

beam beam

in

w=

——

'

-

j

where

inches, d = depth

in feet

of

is

w = load beam

in

in tons, b = breadth of inches, and = length of I

between the s.

Find the safe load for yellow-pine beams 25 and depths from 8 to 18 in.

ft.

long, 4

in.

in breadth, 13.

The number

of cubic feet of air transmitted per minute

in pipes of various diameters is given

q

= number

by q

of cubic feet of air per minute, v

= .327 vd where = velocity of flow 2

,

and d = diameter of pipe in inches. showing the volume of air transmitted in pipes of diameters from 2 to 10 in. with a flow of 12 ft. per second.

in feet per second,

Make a

table

LOGARITHMIC PAPER 14.

The following formula

curvature in paving streets

:

y

is

255

used for computing the surface

=—

aj

2 ,

where x

= horizontal dis-

=

tance in feet from center of street, y vertical distance in inches below grade, a one half the width of the street in feet, b depth of gutter in inches below center of street.

=

=

Pig. 110

Construct a graph to read grade at points

2, 4,

wide, if the gutter

is

6

ft.

15

•

in.

•

off the vertical distances below from the center of a street 60 ft. below the center of the street.

Find the equation connecting x and y when the following corresponding values are given

:

X

15.

V

Suggestion. Locate the points

ting

OX

at

A

and

a line parallel to

AB

is

2

;

YZ

at B.

BA, cutting

and draw a

line

From C on YZ

through them, cut-

directly above

OY at D. OD = 3.5 = m. The

hence the required equation

is

y

=

3.5 z 2 .

A

draw

slope of

APPLIED MATHEMATICS

256 X V

LOGARITHMIC PAPER 23.

The following

table

gives

(F.) of air at different pressures

gain or loss of heat,

t

= absolute

pounds per square inch.

p

the

when

257

absolute temperature

it is

compressed without

temperature

(¥.),

and

p=

258

APPLIED MATHEMATICS •Wire Table

— Copper

Wire

TABLES UNIT EQUIVALENTS Pressure 1 1

pound per square inch pound per square inch

.... ....

1

atmosphere atmosphere atmosphere foot of water at 62° F.

1

inch of mercury at 62° F.

1 1 1

2.042 inches of mercury at 62° F. 2.309 feet of water at 62° F.

.

14.7 pounds per square inch. 30 inches of mercury at 62° F.

.

33.95 feet of water at 62° F. .433 .491

pound per square pound per square

inch. inch.

Length lmil

001 inch.

1

inch

2.54 centimeters.

1

mile

1.609 kilometers.

1

centimeter

1

kilometer

3937 inch. 3280.8 feet.

Area mil

1 circular

.

7854 square mil.

.

1

square mil

1.273 circular mils.

1

square inch square centimeter

645.16 square millimeters.

1

155 square inch.

Volume 1

cubic inch

1

cubic foot

7.48 gallons (liquid, U. S.).

1

pint (liquid, U. S.)

473.18 cubic centimeters.

1

pint (liquid, U. S.)

1

gallon (liquid, U. S.)

1

bushel

1 liter

1 liter

.

28.875 cubic inches. .

.

.

.

231 cubic inches. 2150.4 cubic inches.

centimeter

1 cubic

16.387 cubic centimeters.

.

061 cubic inch. 61.02 cubic inches. 2.113 pints (liquid, U.

259

S.).

APPLIED MATHEMATICS

260 Weight 1

ounce (avoirdupois) ounce (avoirdupois)

1

pound (avoirdupois)

1

ton (2000 pounds)

1

1 cubic

437.5 grains. .

28.35 grams.

.

453.6 grams. .

907.185 kilograms.

.

centimeter of water

1

gram

1

cubic foot of water

.

.

1

cubic inch of water

.

.

1

gallon of water (liquid,

1

gram.

.0353 ounce (avoirdupois).

62.4 pounds. .0361 pounds.

U

8.)

8.345 pounds.

u.)

1

Energy, Work, Heat 1 British

thermal unit (B.

1

British thermal unit

1

British thermal unit

1

power hour horse power hour

1

kilowatt hour

1

kilowatt hour

.

t.

pound water

1° F.

.

.

778 foot pounds.

.

.

746 watt hours.

.293 watt hour.

1 horse

....

...

2544.7 British thermal units.

3412.66 British thermal units.

.

.

.

.

.

1.341 horse

power hours.

Power 1

watt watt

1

horse power

1

horse power

1

1 horse

.... power

44.25 foot pounds per minute. .0569 B.

t.

u.

per minute.

33,000 foot pounds per minute.

.

.

...

....

746 watts per minute. 42.41 B.

t.

u.

per minute.

BIBLIOGRAPHY Pekrt. The Teaching of Mathematics, pp. 101. 1902. The Macmillan Company. 2s. Young. The Teaching of Mathematics, pp. 351. 1907. Longmans, Green

&

Co.

$1.60.

Kent. Mechanical Engineers' Pocket-Book, pp. 1129. 1906. John Wiley

&

Sons.

$5.00.

Seaver. Mathematical Handbook, pp. 365. McGraw-Hill Book Company. $2.50. Slee. The Mechanical Engineer's Reference Book, pp. 859. 1905. J. B. Lippincott Company. $6.00. Trautwine. Engineers' Pocket-Book, pp.

Sons.

1257.

1909.

John Wiley &

$5.00.

Briggs. Eirst Stage Mathematics, pp. 312. 1910. Clive. 2s. Godfrey and Bell. Experimental Mathematics, pp. 64. 1905. Arnold. 2s.

Leighton. Elementary Mathematics, Algebra and Geometry, pp. 296. Blackie.

1907.

2s.

Lodge. Easy Mathematics, Chiefly Arithmetic, pp. 436. Macmillan Company. 4s. 6d. Mair. A School Course of Mathematics, pp. 388. 1907. Press.

Myers.

1906.

The

Clarendon

3s. 6d.

First-Year Mathematics, pp. 365.

1909.

The University

of

Chicago Press. $1.00. Myers. Second^Year Mathematics, pp. 282. 1910. The University of Chicago Press. $1.60. Short and Elson. Secondary School Mathematics, Book I, pp. 182. 1910. $1.00. Book II, pp. 192. 1911. $1.00-. D. C. Heath & Co. Secondary Arithmetic, pp. 221. 1908. Benj. H. Stone-Millis.

A

Sanborn & Co.

75 cents.

Breckenridge, Mersereau, and Moore. Shop Problems matics, pp. 280. 1910. Ginn and Company. $1.00. 261

in

Mathe-

APPLIED MATHEMATICS

262

Castle. Workshop Mathematics, pp. 331. 1900. The Macmillan Company. 65 cents. Castle. Manual of Practical Mathematics, pp. 548. 1904. The Macmillan

Company. $1.50. Consterdine and Barnes. Murray.

Mathematics, pp. 332.

Practical

Cracknell. Practical Mathematics, pp.

& Co. Graham.

1907.

2s. 6d.

378.

1906.

Longmans, Green

$1.10.

Practical Mathematics, pp. 276.

Holton. Shop Mathematics, pp. pany;

212.

1899.

1910.

Arnold.

3s. 6d.

The Taylor-Holden Com-

$1.25.

Jessop. Elements of Applied Mathematics, pp. 344. Sons.

1907.

Geo. Bell

&

4s. 6d.

Knott and Mackat.

Practical Mathematics, pp. 627.

Chambers.

1903.

4s. 6d.

Oliver. Elementary Practical Mathematics, pp. 240. Oliver.

Ormsby.

(Reissue) 1910.

Is. 6d.

Elementary Practical Mathematics, pp. 442.

Chamberlain.

1900.

Spon &

7s. 6d.

Perry. Practical Mathematics, pp. 183. 1907. "Wyman. 9d. Saxelby. Practical Mathematics, pp. 438. 1905. Longmans, Green & Co.

$2.25.

Saxelby. Introductory Practical Mathematics, pp. 220. 1908. Longmans, Green & Co. 80 cents. Stainer. Junior Practical Mathematics, pp. 360. 1906. Geo. Bell & Sons.

3s. 6d.

Starling and Clarke. Preliminary Practical Mathematics, pp. 1904.

Arnold.

Stern and Topham. Practical Mathematics, pp.

&

Sons.

168.

Is. 6d.

376.

1905.

Geo. Bell

2s. 6d.

Gibson. Treatise on Graphs, pp. 181. 1905. The Macmillan Company. $1.00.

Hamilton and Kettle. Graphs and Imaginaries, mans, Green & Co. 50 cents.

pp. 41.

1904.

Long-

Lightfoot. Studies in Graphical Arithmetic, pp. 63. Normal Correspondence College Press. London. Is. 6d. Morgan. Elementary Graphs, pp. 76. 1903. Blackie. Is. 6d. Nipher. Introduction to Graphic Algebra, pp. 60. 1898. Henry Holt and Company. 60 cents.

BIBLIOGRAPHY

263

Phillips and Beebe. Graphic Algebra, pp. 156. and Company. $1.60.

Sohultze. Graphic Algebra, pp.

93.

1908.

1904.

Henry Holt

The Macmillan Company.

80 cents.

Turner.

Graphics applied to Arithmetic, Mensuration, and Statics,

pp. 398.

1908.

The Macmillan Company.

$1.50.

Edser. Measurement and "Weighing, pp. 120. 1899. Chapman & Hall. 2s. 6d.

Graves. Forest Mensuration, pp. 458. 1906. John Wiley & Sons. $4.00. Lambert. Computation and Mensuration, pp. 92. 1907. The Macmillan

Company. 80 cents. Langley. Treatise on Computation, pp.

& Co. $1.00. Larard and Golding. 1907.

Griffin.

184. .1895.

Longmans, Green

Practical Calculations for Engineers, pp. 455.

6s.

Chivers. Elementary Mensuration, pp. 344. 1904. Longmans, Green & Co.

$1.25.

Edwards. Mensuration, pp. 304. 1902. Arnold. 3s. 6d. Eggar. A Manual of Geometry, pp. 325. 1906. The Macmillan Company. 3s. 6d.

Harris. Plane Geometrical Drawing, pp. 270.

1907.

&

Sons.

71.

The

Geo. Bell

2s. 6d.

Myers.

Geometrical Exercises for Algebraic Solution, pp.

University of Chicago Press.

75 cents.

Stone-Millis. Elementary Plane Geometry, pp. 252. 1910. Benj. H. Sanborn & Co. 80 cents. Wright. Exercises in Concrete Geometry, pp. 84. 1906. D. C. Heath & Co. 30 cents.

Bohannan. Plane Trigonometry, pp. 374. 1904. Allyn & Bacon. $2.50. Playne and Eawdry. Plane Trigonometry, pp. 176. 1907. Arnold. 2s. 6d.

Miller. Progressive Problems in Physics, pp. 218. 1909. D. C. Heath

&

Co.

60 cents.

Sanborn. Mechanics Problems, pp. 194. John Wiley & Sons. $1.50. Snyder and Palmer. One Thousand Problems in Physics, pp. 142. 55 cents. 1902. Ginn and Company.

APPLIED MATHEMATICS

264

Atkinson. Electrical and Magnetic Calculations, pp. 310. 1908. D. Van Nostrand Company. |1.50. Hooper and Wells. Electrical Problems, pp. 170. Ginn and Company. $1.25.

James and Sands. Elementary Electrical Longmans, Green & Co. $1.25.

Calculations, pp. 224.

1905.

Shepardson. Electrical Catechism, pp. 417. 1908. McGraw-Hill Book Company. $2.00.

Whittaker. Arithmetic

of Electrical Engineering, pp. 159.

Whittaker.

25 cents.

Behrendsen-GBtting. LehrbuchderMathematik,254S. 1909. Teubner. M. 2.80. Ehrig. Geometrie fur Baugewerkenschulen, Teil I., 138 S. 1909. Leineweber. M. 2.80. Eenkner. Arithmetische Aufgaben, Ausgabe A.,TeilI.,274S. M. 2.20. Salle. Teil II a., 114 S. M. 1.50. Teil 116., 218 S. M. 2.60. Geigenmuller. Hbhere Mathematik, 1. 290 S. 1907. Polytechnische Buchhandlung. M. 6. Muller und Kutnewsky. Sammlung von Aufgaben aus der Arithmetik, Trigonometrie und Stereometrie, Ausgabe B., 2ter Teil, 312 S. 1910. Teubner. M. 3. Schulke. Aufgaben-Sammlung, Teil I., 194 S. 1906. Teubner. M.2.20. Weill. Sammlung Graphischer Aufgaben fur den Gebrauch an hohere Schulen, 64 S. 1909. J. Boltzesche Buchhandlung. M. 1.80.

I.

Four-place Logarithms of Three-Figure Numbers

The Natural

and Cotangents op Angles differing by Ten Minutes, and their

II.

Sines, Cosines, Tangents,

Four-place Logarithms

265

266

1

APPLIED MATHEMATICS

FOUR-PLACE LOGARITHMS 50

267

268 Angle

APPLIED MATHEMATICS

FOUR-PLACE LOGARITHMS Angle

269

270 Angle

APPLIED MATHEMATICS

FOUR-PLACE LOGARITHMS Angle

271

272 Angle

APPLIED MATHEMATICS

;

INDEX Algebra, geometrical exercises for,

Electromotive force, 212 Equations, graphical solution

153

Ammeter, 217 Ampere, 214 Angle functions, 134

Errors,

of,

85

1

Field magnets, 236

Foot pound, 16 Fulcrum, 28

Angles, 54, 134

Approximate number,

2,

120

Archimedes, principle

of,

47

Function, 92 Functionality, 91

Beams, 36 Brake, Prony, 21 friction, 21 British thermal unit, 202

Geometry, algebraic applications,

;

52, 97, 153

;

exercises in solid, nu-

merical, 177; graphical, 186; algeCalipers, vernier, 9

;

micrometer, 12

braic, 190

Graphs, 65, 223 Gravity, 42

Calorie, 202

Characteristic, 121

Cosine, 135 Cosines, law of, 146

Heat, 195

;

linear expansion, 199

Crane, 148

measurement of, 202 mechanical

Density, 42

202 specific, 204 latent, 209 generated by an electric current, 231

;

equivalent

;

;

Digit, 2

Horse power, 17

Division, of approximate numbers, 5;

of,

by logarithms, 124; by

slide

Inequality of numbers, 92

rule, 129

Dynamos, 236;

efficiency of, 237,

Joule, 202

240

Kilowatt, 227

Efficiency, 23, 237, 240 Electricity, 212; units, 213;

work

Kilowatt hour, 227

and power, 227; generation of heat, 231; wiring for light and power, 233 dynamos and motors,

Latent heat, 209

236

Levers, 27

;

273

Leverage, 28

;

;

APPLIED MATHEMATICS

274

Logarithmic paper, 243

Proportion, 110

Logarithms, 120

Protractor, 54

Mantissa, 120

Ratio, 109

"Ready

Mass, 42

Maximum and minimum values,

Measurements, 4 Mechanical advantage, 28 Melting points, 198 Mil, 233 Mil foot, 233 Motors, 236 efficiency of, 237, 240 ;

numbers, 2

by

;

of

Scale,

drawing

by logarithms, 124

Significant figures, 2 Sine, 135 Sines,

law

;

of,

144

Slide rule, 128 of,

65

Tangent, 135

Thermometers, 195 Triangle,

Numbers, exact, 2 approximate, 120

52

approximate

slide rule, 129

;

to,

Series circuit, 216

Squared paper, use

Multiple circuit, 221 Multiplication,

reckoner," 69, 243

93

2,

reference,

of

134

;

of

forces, .147

scale, 91

Variables, 62

Ohm, 213 Ohm's law, 214

Variation, 164

;

inverse, 166

167 Volt, 213

Parallel circuit, 221

Voltmeter, 217

Parallel lines, 59

Parallelogram, 59

Watt, 226

Perpendicular, 55

Watt minute, 226

Power,

Weight, 42 Work, 16, 226

17,

226

Prony brake, 21

.

;

t,

ANNOUNCEMENTS

ELEMENTS OF

APPLIED MATHEMATICS By Herbert E. Cobb, Professor

nmo, This textbook

cloth,

viii

of Mathematics, Lewis Institute, Chicago,

+ 274

111.

$1.00

pages, illustrated,

and manual-training schools is an attempt to relate arithmetic, algebra, geometry, and trigonometry closely to one another and to connect all the mathematics with the work in the shops and laboratories. It replaces the formal, abstract, and purely theoretical portions of algebra and geometry with problems based on the work in the shops and laboratories and with experiments and exercises in the mathematics classroom, where the pupil by measuring and weighing secures his own data for numerical computations and geometrical constructions. Arithmetic is used to check algebraic results, and algebra is made a valuable asset in working out geometrical problems. The problems deal with various phases of real life, and in solving them the pupil finds use for all his mathematics, his physics, and his practical knowledge. The book can be profitably used in conjunction for high schools

with more formal texts,

if

desirable.

VOCATIONAL ALGEBRA By George Wentworth and David Eugene Smith i2mo, "

cloth,

Vocational Algebra "

is

88 pages, for the

illustrated,

boy

50

cents

in the manual-training school

who is not going through high school and has no thought of higher technical training. It presents the simthe ple algebraic conceptions that have a vocational significance or the evening technical class

—

meaning

of algebraic formulas, the solving of simple equations, the use

and the fundamental operations. Any one who has will be able to understand and use the algebra of trade journals, artisans' manuals, and handbooks of business. " Vocational Algebra " may, in many schools, be suitably and profitably introof the negative,

mastered the book

duced

in the eighth grade.

GINN AND COMPANY

Publishers

SHOP PROBLEMS IN MATHEMATICS By William E. Breckenridge, Chairman of the Department of Mathematics, Samuel F. Mersereau, Chairman of the Department of Woodworking, and Charles F. Moore, Chairman of the Department of Metal Working in Stuyvesant High School, New York City

Answer Book furnished on i2mo, 280

Cloth,

THIS

book aims

application

pages, illustrated,

$1.00

to give a thorough training in the mathe-

matical operations that are useful in shop practice, e.g. in

Making and Foundry Work, Forging, and

Carpentry, Pattern

Machine Work, and

much

at the

same time

to impart to the student

information in regard to shops and shop materials.

The

mathematical scope varies from addition of fractions to natural trigonometric functions.

The problems

tions of the processes of

mathematics to the regular work of

the shop. to the

They

more

are graded from simple

difficult

work

their mathematical ability

and

drill

which

facilitate their

All problems are based on actual

the text

Through

will

strengthen

experience.

The

slide rule is

Short methods and checks are emphasized.

illustrations of the

make

board measure

shop work.

Clear explanations of the mechanical

work and

in

exercises of the machine shop.

them students may obtain a double

treated at length.

are practical applica-

common

to

shop

machinery and tools referred to

in

the book an easy one for both student and

teacher to handle. It should

be useful in any school, elementary or advanced,

where there are shops, as a review for supplementary work or as a textbook either in mathematics or shop work.

_

.

GINN AND COMPANY

___ Publishers

:

BETZ AND WEBB

PLANE GEOMETRY i2mo,

'T^HIS *-

is

logical

developments all

that

new

a

cloth,

332

pages,

presentation of geometry along psycho-

as well as logical lines. in

was best

$1.00

It

embodies the

geometry teaching, retaining in the old geometries.

at the

It is the

latest

same time

outgrowth of

the extended experience of two high-school teachers of note,

and

a fresh, sane, teachable textbook that

is

will

be welcomed

by teachers the country over who have been waiting

for just

such a presentation of the subject.

Some 1.

A

of the features

preliminary course precedes the demonstrative course,

vitaliz-

ing definitions by abundant illustration and discussion, cultivating skill in the use of ruler and com through interesting drawing exercises,

and presenting exercises requiring simple reasoning and inference. .2. The topical plan is followed. Difficult topics are approached by

means

of a preliminary discussion.

3.

Hypothetical figures are avoided.

4. 5.

Area precedes similarity. The incommensurable case

6.

The theory

of limits

is

is

made

made unnecessary.

optional.

It is

preceded by an

alter-

native informal discussion. 7.

The

different types of exercises

— constructions,

computations,

—

and original theorems receive approximately equal attention. 8. The applied problems are numerous but not excessive in number.

The aim throughout

is to

make

the pupil independent

of the textbook

130

GINN AND COMPANY

Publishers

HAWKES, LUBY, AND TOUTON'S ALGEBRAS By Herbert E. Hawkes, Professor of Mathematics in Columbia University, William A. Luby, Head of the Department of Mathematics, Central High School, Kansas City, Mo., and Frank C. Touton, Principal of Central High School, St. Joseph, Mo.

FIRST COURSE IN ALGEBRA trated,

i2mo, cloth,

vii

+

334 pages,

illus-

JjSi.oo.

SECOND COURSE IN ALGEBRA

121™, cloth,

viii

+ 264

pages,

illustrated, 75 cents.

COMPLETE SCHOOL ALGEBRA

iano,

cloth, *i

+

507 pages,

illustrated, JS1.25.

Hawkes, THE ment

Luby, and Touton Algebras offer a fresh treatcombining the best in the old methods of teaching algebra with what is most valuable in recent developments. The authors' unhackneyed and vital manner of presenting the sub.

of the subject,

ject

makes a sure appeal

to the interest of the student, while their

genuine respect for mathematical thoroughness and accuracy gives the teacher confidence in their work.

Among

the distinctive features of these algebras are the correlaand physics; the liberal use of illustrative material, such as brief biographical sketches of the mathematicians who have contributed materially to the science early and extended work with graphs and the introduction of numerous " thinkable " problems. Prominence is given the equation throughout, and the habit of checking results is constantly encouraged. Thoroughness is assured by frequent short reviews. The aim has been to treat in a clear, practical, and attractive manner those topics selected as necessary for the best secondary schools. The authors have sought to prepare a text that will lead the student to think clearly as well as to acquire the necessary facility on the technical side of algebra. The books offer a course readily adaptable to the varying conditions in different schools the " Complete School Algebra " comprising a one-book course with material sufficient for at least one and one-half year's work, and the " First Course" and " Second Course " providing the same material, but slightly tion of algebra with arithmetic, geometry,

;

;

—

expanded, in a two-book course.

GINN AND COMPANY

Publishers

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