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2. The hypotenuse of a right-angled triangle is 45 inches, and the base 27 inches: what is the perpendicular ?

3. The perpendicular of a 'right-angled triangle is 57 feet, and the hypotenuse 95 feet : what is the base ?

4. Two vessels sail from the same port, one south 4 miles an hour, and the other east 5 miles an hour : how far apart are they at the end of 12 hours ?

5. If a ladder 48 feet long touches the side of a building at a point 36 feet above the ground, how far is the bottom of the ladder from the base of the building?

6. The height of a tree on the bank of a river is 96 feet, and a line reaching from its top to the opposite bank is 120 feet: what is the width of the river ?

7. A flagstaff 80 feet high casts a shadow 60 feet in length : what is the distance from the top of the staff to the extremity of the shadow ?

8. A tree was broken 36 feet from the bottom, and fell so that the end struck 42 feet from the foot: what was the height of the tree?

9. What is the distance from a lower corner to the upper opposite corner of a room 40 ft. long, 30 ft. wide and 20 ft. high ?

10. Illustrate Art. 437 by an original problem.

ART. 438.—Comparison of Similar Figures.

Principles.-1. The areas of similar figures are to each other as the squares of their like dimensions.

2. The dimensions of similar figures are to each other as the square roots of their areas.

1. One side of a square is 15 feet, and one side of another square is 20 feet : how many times the area of the first is the area of the second ?

2. A certain square field contains 144 rods, and another similar field contains 1,296 rods : how many times the length of the side of the first is the side of the second ?

3. If a pipe 21 inches in diameter can empty a tank in 1 hour, how long will it take a pipe 2 inches in diameter to empty it ?

4. The area of a circle, whose diameter is 20 feet, is 314.16 square feet: what is the diameter of a circle whose area is 157.08 square feet ?

5. If the area of a circle, whose diameter is 20 feet, is 314.16 square feet, what is the diameter of a circle whose area is 2,827.44 square feet ?

6. What is the length of the edge of a cubical block whose entire surface is 4,704 square feet?

7. Find the edge of a cube whose entire surface is 33,750 square feet.

8. Illustrate Art. 438 by an original problem.

Cube Root.

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= 30

WRITTEN EXERCISES. Multiplying the square of any number, as 35, by the number itself gives the cube of the number. Doing this in the manner already explained, 352 : 302 + 2 (30 x 5) + 52 35

+ 5
303 + 2 (302 x 5) + (30 x 52)

(302 x 5) + 2 (30 x 52) + 53 353 = 303 + 3 (302 x 5) + 3 (30 x 52) + 53 = 42,875.

Hence, the cube of any number consisting of tens and units equals the cube of the tens, plus three times the square of the tens multiplied by the units, plus three times the tens multiplied by the square of the units, plus the cube of the units.

=

23 = 8.

From the cubes in the margin we 53 = 125. may infer that the cube of any num123 = 1728. ber contains three times as many fig

993 970299. ures as the number itself, three times as many less one, or three times as many less two. Hence,

If a number be separated into periods of three figures each, beginning at the right, there will be as many periods as there are figures in its cube root.

The left hand period may contain one, two, or three figures.

1. Extract the cube root of 42,875.

Analysis.-Separating Process.

the number into periods 42'875(35 in accordance with the 27

principle developed above, Tens? x 3 = 270015875

we learn that there will Tens x units x 3 =

450

be two figures in the cube Units 2 25

root. The first figure of 3175 15875 the root is found by tak

ing the cube root of the greatest cube in the left hand period. This root is 3, and we write it as the tens' figure of the root. Placing its cube under the left hand period and subtracting, we have a remainder of 15, to which the next period is annexed to form a new dividend. From the composition of the cube of 35, as explained above, it is clear that this dividend, 15,875, is what is left after subtracting the cube of the tens in 35 from the cube of 35. But tens? x 3 x units or 2,700 times the units, forms almost all this dividend ; hence, if we divide by 3 times the square of the tens, or 2,700, and remember that the quotient is likely to be too large, since the divisor is too small, we shall obtain the units' figure. The trial divisor 2,700 is now increased by 3 times the tens x units, and by the square of the units, and the sum is multiplied by the units. The product 15,875 being equal to the dividend, the operation is complete

ART. 439.—Rule for extracting the cube root of a number.Beginning at the right, separate the number into periods of three figures each.

Find the greatest cube contained in the left hand period, and place its cube root at the right hand for the first term of the root. Subtract its cube from the left hand period, and annex the next period to the remainder for a dividend.

Take three times the square of the first term of the root regarded as tens, for a trial divisor, divide the dividend by it, and the quotient will be the second term of the root. Add to the trial divisor three times the product of the first part of the root, considered as tens, by the last term of the root; also the square of the last term. The sum is the complete divisor.

Multiply the complete divisor by the second term of the root; subtract this sum from the dividend, annex the next period to the remainder for the second dividend, and proceed as before.

NOTE.—Should the trial divisor at any time not be contained in the dividend, write a cipher in the root, annex two ciphers to the trial divisor, bring down the next period, and proceed as before.

Extract the cube root of

3. 12167.
4. 15625.
5. 35937.
6. 970299.
17. 2571353.
8. 34328125.
9. 131096512.

10. 633839.779.
11. .000512.
12. 49:37:
13. 3437
14. 2.
15. 3.
16. 6.

ART. 440.-Applications of Cube Root.

1. A cubical box contains 474,552 cubic inches : what is the length of one of its sides?

2. How many square feet in the surface of a cube whose volume is 389,017 cubic inches ?

3. What is the area of one of the sides of a cube which contains 185,193 cubic inches?

4. What is the depth of a cubical cistern whose capacity is 6,591 cubic feet?

5. Find the side of a cubical vat whose capacity is 5,545,233 cubic inches.

6. A certain apartment of cubical form contains 1,953,125 cubic feet of air : what is the height of the room ?

7. To excavate a cubical cellar, 450 cubic yards and 17 cubic feet were taken out: what was the length of one side of the cellar ?

8. What is the side of a cube of stone equal in volume to a monument 8 feet square and 81 feet high ?

9. What is the length of a cubical pile of stone equal in volume to a rectangular pile whose length is 64 feet, breadth 24 feet, height 9 feet?

10. What is the side of a cube equal to a pile of wood 1,024 feet long, 4 feet wide, 28 feet high ?

11. A rank of wood 120 feet long, 8 feet wide, and 15 feet high, is placed in a cubical pile : how high is it?

12. Two blocks of granite are together equal to another of cubical form; the first is 18 feet long, 9 feet wide and 6 feet deep; the second is 33 feet long, 18 feet wide, and 9 feet deep : what is the side of the cube ?

13. A certain room is twice as long as high, and the width is equal to the height: what is the height, if the capacity of the room is 3,456 cubic feet?

14. Illustrate Art. 440 by an original problem.,

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