16. How many solid or cubic feet, in a piece of scant ng 12 feet long, 7 inches wide, and 8 inches deep. Remark.—If this piece of scantling were 1 foot wide and 1 foot deep, it would then contain 12 cubic feet, and if it were 1 foot wide and only 8 inches deep, it would contain but of 12 cubic feet, (8 inches being of a foot,)-now since it would contain of 12 feet, by being 8 inches deep and 1 foot wide, it will contain but as much by being 71⁄2 inches wide, (as 71⁄2 inches is of a foot,) hence, we state the question, Again, suppose we look at it thus, a cubic foot being 12 inches long, 12 inches wide, and 12 inches deep, contains 1728 cubic inches, which is the result of multiplying together the numbers representing the inches, in the length, width, and depth or thickness. If, in like manner, we multiply to gether the numbers representing the inches in the length, width, and depth or thickness of the scantling, we will have as the result, the number of cubic inches it contains. Thus, 12 ft.=144 in. ×74×8=8640 cubic inches, which is 5 times 1728, therefore the scantling contains 5 cubic feet. Hence, if the numbers representing the length, width, and depth, be multiplied together, the result will be the cubic contents in the same denomination as the factors. 17. How many cubic feet does a slab of marble, 4 feet 4 inches long, 3 feet 7 inches wide, and 1 foot 4 inches thick, contain? 8 35 12 43 34 72 1505(2025 cubic ft. Ans. 18. What is a pile of wood 28 feet long, 8 feet wide, and 7 feet 6 inches high, worth, at $4.00 a cord 128 2 15 A cord 128 ft. A Ans. $52.50 19. How many cords of wood feet wide, and 12 feet high? in a pile 47 feet long, 32 Ans. 141 cords. (0.) 20. What is the value of a pile of wood 97 feet 9 inches long, 24 feet wide, and 5 feet 4 inches high, at the rate of $2.50 a cord? Ans. $244.37. (0.) 21. A tanner has a bark house 96 feet long, 37 feet 6 inches wide, and 18 feet 8 inches high to the beam, to which he wished to fill it with bark, at an average price of $7.25 a cord, what sum will it cost? Ans. $3806.25. (8.) 22. If a pile of stones be 54 feet long, its average width 16 feet 6 inches, and height 7 feet 3 inches; how many perches does it contain, a perch of stone being 244 cubic feet. Ans. 261 perches. (0.) 23. What is the value of a pile of stone, the average length, width, and height, being 25 feet, 12 feet, and 5 feet, in order, at 75 cents a perch? Ans. $52.08. (0.) 24. A man has a cellar 28 feet long, 17 feet 6 inches wide, and 7 feet 4 inches deep, to be walled with stone, the wall to be 15 inches thick; what will the stone cost at 72 cents a perch? Ans. $22.963. (4.) NOTE. You must deduct 4 times the thickness of the wall from twice the sum of the length and width, for if the side walls are supposed to extend the whole length of the cellar, the length of each end wall will be twice the thickness of the wall, less than the width of the cellar. 25. A agrees to dig a cellar for B, 34 feet 8 inches long, 27 feet 6 inches wide, and 6 feet 9 inches deep, at 9 cents per cubic yard, what will it amount to? Ans. $21.45. (0.) 26. A farmer has a bin 19 feet 6 inches long, 3 feet 10 inches wide, and 4 feet 8 inches deep, filled with wheat, which he wishes to sell at $1.08 a bushel, what is the whole worth at this rate? Ans. $302.73. (18.) NOTE. The standard gallon, dry measure, contains 2684 cubic inches, and of course the bushel 8 times 2684 cubic inches. 27. How many hogsheads of water will a vat 9 feet 9 inches deep, 7 feet 7 inches long, and 4 feet 6 inches wide hold? The gallon, wine measure (by which water is measured) contains 231 cubic inches. Ans. 3932 hogsheads. (11.) 28. A bushel measure is 18 inches in diameter, and 8 inches deep, how many cubic inches does it contain? NOTE. In order to solve this question, take, instead of the bushel measure, a box 8 inches deep, and 18 inches across each side, measured in the clear, and suppose the annexed figures, one to represent the bottom of the box, and the other that of the measure; then, if you multiply together the length width, (each 18 inches) and depth of the box, the result will be the number of cubic inches contained by the box; but it is obvious that the bushel measure will not contain so much, since a quantity equal to that contained in the four corners of the box will remain after the bushel is filled. Now it has been ascertained that the area of a circle, whose diameter is 1 will be .7854, a little more than , whilst the area of a square whose side is foot or 1 inch, &c., will be 1 cubic foot, 1 cubic inch, &c. Hence if you multiply the contents of the box by the decimal .7854, you find the cubic inches in a bushel to be 2150.4252. 29. How many gallons of water will a tub that is 3 feet 8 inches in diameter, and 2 feet 3 inches deep hold? Ans. 177.7248 gals. (10.) 30. How many gallons of water will a cistern 8 feet 3 inches deep, and 3 feet 6 inches in diameter hold? Ans. 593.7624 gals. (3.) INVOLUTION. Involution is the raising of powers from any given number as a root. A power is the number resulting from the multiplication of any given number called a root, a certain number of times in regular succession by itself; for example let-3 be multiplied into itself. Thus, 31-3 the root or first power. 32-3x3=9 is the 2d power or square. 35=3x3x3x3x3=243 is the 5th power of 3. The number denoting the power is called the index or exponent of the power-thus-35 denotes the fifth power of 3, and 5 is the index or exponent. NOTE.-The square of the second power will be the fourth power. 21. Multiply the second power of 4 by the second power of 4-thus, 4 x4°=4=(4×4)x(4x4)=256. 22. Multiply the third power of 4 by the second power of 4-thus 43x4o=45=(4×4×4)x(4×4)==1024. 23. Multiply the fourth power of 4 by the fourth power of 4-thus, 4x4-48=(4×4x4x4)x(4x4x4x4)=65536. 24. Multiply the fifth power of 3 by the fourth power of 3, thus, 35×34=39=(3×3×3×3×3)×(3×3×3×3)=19683. Hence, we see that the second power multiplied by the second power produces the fourth power; the third power multiplied by the third power produces the sixth power; the fourth power multiplied by the fourth power produces the eighth power, &c. &c. Let the answers to as many of the following questions as possible be given by a purely mental operation. 25. What is the second power of 29, of 14, of 13, of 47, of 16? Y * 26. What is the third power of 3, of 2, of 4, of 5, of 6, of 7, of 8? 27. What is the fourth power of 1, of 2, of 3, of 4, of 5, of 6, of 7, of 8? 28. What is the fifth power of 1, of 2, of 3, of 4, of 5, of 6, of 7, of 8, of 9? 29. What is the sixth power of 1, of 2, of 3, of 4, of 5, of 6, of 7, of 8, of 9? 30. What is the seventh power of 1, of 2, of 3, of 4, of 5, of 6, of 7? 31. Multiply the second power of 3 by the fourth power of 3, 42 by 43, 2a by 2*, 53 by 55, 6o by 63, 83 by 8o, 93 by 9o 74 by 73. 32. If the third power of 4 be multiplied by the fourth power of 4, what power of 4 will the product be? Ans. The seventh power of 4, that is, the sum of the exponents expresses the power. 33. What power of 6 will 61×65 give? 34. What power of 8 is 83×85 equal to ? Ans. 69. Ans. 88. 35. What power of 7 is 75x75, of 4 is 47x46, of 9 is 98×94, of 5 is 57x53? 36. What power of 3 is 33×31×3o, of 8 is 8a×83×83, of 2 is 27×28? Since the sum of the exponents expresses the power resulting from the multiplication of different powers of a number together, and division is the opposite of multiplication, the difference of the exponents will express the power resulting from the division of one power by another. Thus, if we divide the fifth power of 8 by the third power of 8 we get the second power. 37. Divide 96 by 93, 87 by 84, 68 by 65. 38. Divide 174 by 172, 145 by 143, 136 by 13o. 39. Divide 87×84 by 85, 48×43×4a by 45. 40. Divide 4 by 48, 29 by 25, 37 by 37. |