7. There is a triangle containing 72 square rods, and one of its sides measures 12 rods; what is the area of a similar triangle whose corresponding side measures 8 rods? Ans. 32 rods. 8. A gentleman has a park, in the form of a right-angled triangle, containing 950 square rods, the longest side or hypothenuse of which is 45 rods. He wishes to lay out another in the same form, with a hypothenuse the length of the first; required the area. Ans. 105.55+ square rods. 9. If a cylinder 6 inches in diameter contain 1.178+ cubic feet, how many cubic feet will a cylinder of the same length contain that is 9 inches in diameter ? Ans. 2.65+ feet. 10. If a pipe 2 inches in diameter will fill a cistern in 201 minutes, how long would it take a pipe that is 3 inches in diameter? Ans. 9 minutes. 11. A tube of an inch in diameter will empty a cistern in 50 minutes ; required the time it will empty the cistern, when there is another pipe running into it of an inch in diameter. Ans. 624 minutes. 293. To find the SIDE of a square that can be inscribed in a circle of a given diameter. A square is said to be inscribed in a circle when each of its angles or corners touches the circumfer It may be conceived to be composed of two right-angled triangles, the base and perpendicular of each being equal, and their hypothenuse the diameter of the circle, as seen in the diagram. Hence the RULE. Extract the square root of half the square of the diameter, and it is the side of the inscribed square. HYPOTH. ence. EXAMPLES FOR PRACTICE. 1. What is the length of one side of a square that can be inscribed in a circle, whose diameter is 12 feet? Ans. 8.48+ ft. 2. How large a square stick may be hewn from a round one which is 30 inches in diameter ? Ans. 21.2+- inches square. 3. A has a cylinder of lignum-vitæ, 194 inches long and 11 inches in diameter; how large a square ruler may be made from it? Ans. 1.06+ inches square. 293. When is a square said to be inscribed in a circle? Of what may the inscribed square be conceived to be composed? What part of the circle is the hypothenuse of the two triangles ? The rule for finding the side of the inscribed square ? EXTRACTION OF THE CUBE ROOT. 294. The Cube Root is the root of a third power. It is so called, because the cube or third power of any number represents the contents of a cubic body, of which the cube root is one of its sides. 295. To extract the cube root of a number, is to find such a . factor as, when multiplied into itself twice, will produce the given number; or it is to resolve the number into three equal factors. Roots of the first ten integers and their cubes are, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 8, 27, 64, 125, 216, 343, 512, 729, 1000. It will be observed that the cube or third power of each of the numbers contains three times as many figures as the root, or three times as many wanting one, or two at most. Hence, to determine the number · of figures in the cube root of a given number, Beginning at the right, point it off into as many periods as possible of three figures each, and there will be as many figures in the root as there are periods. Ex. 1. I have 17576 cubical blocks of marble, which measure one foot on each side ; what will be the length of one of the sides of a cubical pile which may be formed of them? Ans. 26 feet. The number of blocks or feet on a 1 3 5 7 5 ( 26, Roots side will be equal to 8 the cube root of 22 X 300–1200) 9 5 7 6, 1st dividend. 17576. (Art. 294.) Beginning at the 7 2 0 0, 1st addition. right hand, we point 62 X 2 X 30 2 1 6 0, 2d addition. off the number into 63 2 1 6, 3d addition. periods, by placing a point over the right9 5 7 6, Subtrahend. hand figure of each period. We then find the greatest cube number in the left-hand period, 17 (thousands), to be 8 (thousands), and its root 2, which we place in the quotient or OPERATION. 294. What is the cube root, and why so called ? — 295. What is meant by extracting the cube root? How many more figures in the cube of any number than in the root? How do you ascertain the number of figures in the cube root of any number? What is found by extracting the cube root of the number in the example? What is first done after separating the number into periods ? 20 20 20 20 root. As 2 is in the place of tens, because there is to be another figure in the root, its value is 20, and it represents the side of a cube (Fig. 1) the contents of which are 8000 cubic feet; thus, 20 X 20 X 20=8000. Fig. 1. We now subtract the cube of 2 (tens) 8 (thousands) from the first period, 17 (thousands), and have 9. (thousand) feet remaining, which, being increased by the next period, makes 9576 cubic feet. This must be added to three sides of the cube, Fig. 1, in order that it may remain a cube. To do this, we must find the superficial contents of the three sides of the cube, to which the additions are to be made. Now, since one side is 2 (tens) or 20 feet square, its superficial contents will be 20 x 20 = 400 square feet, and this multiplied by three will be the superficial contents of three sides; thus, 20 X 20 X 3 1200, or, which is the same thing, we multiply the square of the quotient figure, or root, by 300; thus, 22 x 300 1200 square feet. Making this number a divisor, we divide the dividend 9576 by Fig. 2. it, and obtain 6, which we place in the root. This 6 represents the thickness of each of the three additions to be made to the cube, and their superficial contents being multiplied by it, we have 1200 X 6 7200 cubic feet for the contents of the three additions, A, B, and C, as seen in Fig. 2. Having made these additions to the cube, we find that there are three other deficiencies, n n, o 0, and rr, the length of •which is equal to one side of the additions, 2 (tens), or 20 feet; and their breadth and thickness, 6 feet, equal to the thickness of the additions. Therefore, to find the solid contents of the additions, necessary to supply these deficiencies, we multiply the product of their length, breadth, and thickness, by the number of additions ; thus, 6 X 6 X 20 X 3 = 2160, or, which is the same thing, we multiply the square of the last quotient figure by the former figure of the root, and that product by 30; thus, 6* X 2 X 30 2160 cubic feet for the contents of the additions S s, u U, and v v, as seen in Fig. 3. 295. What is done with this greatest cube number, and what part of Fig. I does it represent? What is done with the root? What is its value, and what part of the figure does it represent? How are the cubical contents of the figure found? What constitutes the remainder after subtracting the cube number from the left-hand period? To how many sides of the cube must this remainder be added ? How do you find the divisor? What parts of the figure does it represent? How do you obtain the last figure of the root ? What part of Fig. 2 does it represent? What parts of the figure does the product represent? What three other deficiencies in the figure ? 20 6 These additions being made to the cube, we still observe another deficiency of the cubical space x x x, the length, breadth, and thickness of which are each equal to the 6 thickness of the other additions, or 6 feet. Therefore, we find the contents of the addition necessary to supply this deficiency by multiplying its length, breadth, and thickness together, or cubing the last figure of the root; thus 6 X 6 X 6= 216 cubic feet for the contents of the ddition z z , as seen in Fig. 4. If we add together the several additions that have been made, thus, 7200 + 2160 + 216 9576, we obtain the number of cubic feet remaining after subtracting the first cube, which, being subtracted from the dividend in the operation, leaves no remainder. Hence the cubical pile formed is 26 feet on each side ; since 26 x 26 x 26 26 17576, the given number of blocks, and the sum of the several parts of Fig. 4. Thus, 8000 + 7200+2160+216 = 17576. RULE. Separate the given number into as many periods as possible of three figures each, by placing a point over the unit figure, and every third figure beyond the place of units. Find the greatest cube in the left-hand period, and place its root on the right. Subtract the cube, thus found, from this period, and to the remainder bring down the next period for a dividend. Multiply the square of the root already found by 300 for a divisor, by which divide the dividend, and place the quotient, usually diminished by one or two units, for the next figure of the root. Multiply the divisor by the last figure of the root, and write the product under the dividend; then multiply the square of the last figure of the root by its former figure or figures, and this product by 30, and place the product under the last; under all set the cube of the last figure of the root, and call their sum the subtrahend. ? 295. How do you find their contents ? What parts of Fig. 3 does the product represent What other deficiency do you observe? To what are its length, breadth, and thickness equal ? How do you find its contents ? What part of Fig. 4 does it represent? The rule for extracting the cube root? Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before ; and so on, till the whole is completed. NOTE 1. - When the number of the figures in the given number is not divisible exactly by 3, the period on the left will contain less than 3 figures. Note 2. — The observations made in Notes 2, 3, and 4, under square root, are equally applicable to the cube root, except in pointing off decimals each period must contain three figures, and two ciphers must be placed at the right of the divisor when it is not contained in the dividend. EXAMPLES FOR PRACTICE. 1. What is the cube root of 78402752 ? Ans. 428. OPERATION. 78 4 0 2 7 5 2 ( 4 2 8, Root. 64 . 8 100 88 1st subtrahend. 4 2 3 3 6 0 0 5 1 2 2. What is the cube root of 74088 ? Ans. 42. Ans. 57. Ans. 432. Ans. 561. Ans. 636. Ans. 123. Ans. 1234. Ans. 346.7. Ans. 78.51. Ans. 2.012. Ans. .423. 295. How many ciphers must be placed at the right of the divisor when it is not contained in the dividend ? |