3. When a shadow 6 ft. 8 in. is cast by an upright post 5 ft. 10 in. high, what is the height of a spire casting a shadow 216 feet? 4. In walking 3600 feet, a man takes steps of 2 ft. 6 in. How many feet would he walk with the same number of steps, if each step was 2 ft. 4 in.? 5. If a cistern 171⁄2 ft. long, 54 ft. wide, and 13 ft. deep, hold 273 barrels, how many barrels does a cistern hold that is 16 ft. long, 7 ft. wide, and 71⁄2 ft. deep? 6. If 12 men, working 9 hours a day for 155 days, were able to finish of certain work, how many men may be discharged that the work may be completed in 15 days more, at 7 hours work a day? 7. If 248 men, in 63 days of 10 hours each, dig a trench of 7 degrees of hardness, 232 yds. long, 3 yds. wide and 2 yds. deep, in how many days of 9 hours each can 155 men dig a trench of 4 degrees of hardness, 387 yds. long, 10 yds. wide, and 14 yds. deep? 8. If 24 boys can pick 18 cases of berries, each contain ing 12 boxes, in 1 hour, how many such boys could pick 6 cases, each containing 12 such boxes, in 30 minutes? 9. If a block of granite 8 ft. long, 5 ft. wide, 21 ft. thick, weigh 16,335 lbs., what is the weight of a granite column 20 ft. high, 23 ft. square? 10. Divide $960 between A and B so that A shall have 5 times as much money as B. 11. C's money is 1 times as much as D's. They have together $315; how many dollars has each? 12. Divide 462 into four parts that shall be in the ratio of,,, and . f. CHAPTER XIV. INVOLUTION AND EVOLUTION. INVOLUTION. Article 530. A Power of a number is the product of that number used as a factor a certain number of times. Thus, 4 is a power of 2, since 4=2X2. So 16 is a power of 2, since 16=2×2×2×2. 531. Powers are named from the number of times the given number is used as a factor. The First Power of a number is the number itself. The Second Power of a number is the product of that number used twice as a factor. The Third Power of a number is the product of that number used three times as a factor. The fourth power of 4 is 4X4X4X4=256. 532. The second power of a number is usually called its square. The third power of a number is frequently called its cube. See the tables of square and cubic measure. Arts. 297 to 300. 533. The Exponent of a power shows how many times the given number has been used as a factor. The exponent is a small figure written to the right of and above the number representing a power. Thus,5o=5×5: 8o=8×8×8: 5o is read the second power of 5, or, 5 squared. 8o is read the third power of 8, or, the cube of 8; or, 8 cubed. 534. Involution is the process of finding the power of a num ber. 1. Find the fourth power of 12. SOLUTION. 124=12×12×12×12=20736. 2. Find the fifth power of .03. SOLUTION. .035=.03.03X.03.03.03=.0000000243. 3. Find the cube of . SOLUTION. ()*={×3×?=127. Hence, 535. To find a required power of a number: Use the given number as a factor as many times as there are units in the exponent of the power required. Find the following required powers: 536. A Root of a number is one of its equal factors. 16 2 × 2 × 2 × 2, 2 is a root of 4, of 8, and of 16. 537. A Root is named from the number of times it is used as factor. Thus, 2 is the second, or square root of 4; the third, or cube root of 8; the fourth root of 16. 538. A number is its own first root as well as its own first power. The Second, or Square Root of a number is one of its two equal factors. The Third, or Cube Root, of a number is one of its three equal factors, etc. 539. Powers and Roots are correlative. Thus, 8 is the third power of 2; and 2 is the third root of 8. 540. The sign V is called the Radical Sign, and is used to mdicate a root. Thus, 81, or 81, means the square root of 81, 125 means the cube root of 125. The figure on the left of the radical sign is the Index of the root, and shows which root is meant. The Index is usually omitted when the radical indicates the square root. of 8. 1 541. A Perfect Power is a number whose exact root may be obtained. Thus, 27 is a perfect cube, since its cube root, 3, can be obtained. But 27 is not a perfect square. 542. Evolution is the process of obtaining the root of a num ber. 543. Table of a few roots and powers. Roots. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. SQUARE ROOT. 544. The square of any number represented by two or more figures, as 46, may be thus obtained : Every perfect square consists of three parts, as here shown, viz. : 1. The square of the tens; plus 2. Twoice the product of the tens and units; plus 3. The square of the units. Or, more briefly, using t for tens, and u for units : This is illustrated by the following diagram, in which the square A is the square of the tens, (40), each of of the two equal rectangles B and C, is the product of the tens and units, (40×6) and the square D is the square of the units (62). The entire square is therefore equal to 40+2 (40×6)+62, or, in general, t2+2 (txu)+u2. In finding the square root of such a number, it is desirable to retrace the steps taken above. Thus, 1. Find the square root of 2116. PROCESS. 16 86)516 516 ANALYSIS. 1. Separate the number into periods of two figures each, beginning with the units figure. 2. Find the greatest square in the left hand period. The greatest square in 21 is 16; its root, 4, is the tens' figure of the root sought. 3. From the left hand period subtract the square 16, and annex the right hand period. 4. The number thus formed, 516, contains the other two parts of the square, viz.: 2(txu)+u2. Divide 51 of the 516 by 2 t, that is, by 8. The quotient, 6, is probably the units' figure of the root. 5. Find the product of 2 t+u, that is, of 86, multiplied by u, that is, by 6. This product, which is 2(txu)+u2, is 516. Therefore the square root of 2116 is 46. 2. Find the square root of 2481. PROCESS. ANALYSIS. This number is not a 24.81 (49.44+ perfect square, but the process is the 16 89)881 801 1784)8000 17884)86400 same as before. 1. Subtract 16, the greatest square in 24 from 24, writing 4, the square root of 16, as the first figure of the root, and annex the next period, 81, to the remain. der 8, making 881. 2. Double the root, 4, making 8; 8 is contained 9 times in the 88 of 881; therefore 9 is probably the second figure of the root. 3. Write the 9 as the second figure of the root, and as the second figure in the trial divisor, 89. |