COMPARISON OF SIMILAR SOLIDS. Art. 262. The contents of similar solids are to each other as the cubes of their corresponding dimensions. 65. There are 2 cubes; the side of one of them is 2 inches, the side of the other is 4 inches. How many times greater is the cube whose side is 4 inches, than the cube whose side is 2 inches? angle, is equal to the sum of the squares described upon the base and perpendicular. Hence, when two sides of any right-angled triangle are given, we can easily find the third side. When the base and perpendicular are given, we can find the hypothenuse by extracting the square root of the sum of the squares of the base and perpendic 2 X2 X2=8, the number of cubic inches in the smaller cube.ular. When the hypothenuse and 4 X 4X4 64, the number of cubic inches in the greater cube; either of the other sides are given, and 64 inches are 8 times 8 inches. we can find the third side by exHence, we perceive that when tracting the square root of the difthe corresponding dimensions of ference of the squares of the hytwo similar solids are to each oth-pothenuse and given side. er as 2 to 1, the contents are as 8 to 1. When the corresponding dimensions are as 3 to 1, the con tents are as 27 to 1. When the corresponding dimensions are as 4 to 1, the contents are as 64 to 1, &c. When the corresponding dimensions are as 2 to 3, the contents are as 8 to 27. When the corresponding mensions are as 5 to 6, the tents are as 125 to 216, &c. 68. There is a right-angled triangle; its base is 8 feet, and its perpendicular 6 feet. What is the length of the hypothenuse? 69. If the hypothenuse of a right-angled triangle be 20 yards, and the perpendicular 12 yards, what must be the length of the base? 70. If the hypothenuse of a right-angled triangle be 10 rods, di-and the base 8 rods, what must be con- the length of the perpendicular? 66. There are 2 cannon balls; one is 3 inches in diameter, and the other 5 inches in diameter. What is the ratio of the weight of the smaller ball to the weight of the larger? 67. There are two similar square pyramids; the corresponding sides at the base are as 10 to 12. What is their comparative solidity? 71. If the area of a right-angled triangle, whose base is 8 rods, be 24 square rods, what must be the area of a similar triangle, whose base measures 4 rods? 72. If the hypothenuse of a right-angled triangle be 20 feet, and the hypothenuse of a similar right-angled triangle be 12 feet, what is the ratio of the area of the smaller triangle to the area of the greater? 73. If the base of a right-angled Art. 263. We have triangle be 16 feet, and the pershown by a diagram in Art. 189, pendicular 12 feet, what is the that the square described upon the length of the hypothenuse? what hypothenuse of a right-angled tri-is the area of the triangle? Art. 264. 74. Purchased the following articles of furniture: 1 book-case, @ $25; 2 bureaus, @ $12.75; 18 chairs, @ $1.621; 4 bedsteads, @ $7.50; 25 yards of carpeting, @ $1.12; and 63 yards of sheeting, @ 15 cents. What was the amount of the bill? 75. Sold 6 pieces of linen, No. 1, measuring 25 yards, @ $1.12; No. 2, 24 yds., @ 75 cents; No. 3, 254 yds., @ 62 cents; No 4, 25 yds., @ 50 cents; No. 5, 24 yds., @ 40 cents; No. 6, 214 yds., @37 cents. What was the amount of the bill? 76. Purchased 6 lbs. of green tea, @ 65 cents; 10 lbs. of black tea, @ 35 cents; 14 lbs. of loaf sugar, @ 15 cents; 28 lbs. of brown sugar, @ 8 cents; and 75 lbs. of coffee, @ 10 cents. What was the amount of the bill? 77. Bought a load of wood, measuring 8 feet in length, 3 feet 9 inches in width, and 4 feet 6 inches in height; what did it cost, at 92 cents a foot? 78. What will be the whole cost of a cargo of coal delivered in Boston, containing 325 tons, which cost in Philadelphia, $3.75 a ton, freight $1.75 a ton, other charges 25 cents a ton? 79. What number of feet of boards will be required to make a fence 6 feet in height, which shall enclose a rectangular piece of land 16 rods in length, and 12 rods in breadth, allowing nothing for waste? What will they cost at $17.50 a thousand? REMARK. When an article of merchandise is sold at a specified number of dollars per thousand feet, the price of 1 foot will be as many mills as there are dollars in the price per thousand. When an article is sold by the hundred, the price of a unit will be as many cents as there are dollars in the price per hundred. 80. What number of bricks, 8 inches long, 4 inches wide, and 2 inches thick, will be required to build a wall 6 feet in height, and 12 inches thick, which shall enclose the above piece of land? What will the bricks cost, at $7.50 a thousand? 81. Divide $408 among A, B, and C ; and give B three times as much as A, and C four times as much as B. What number of dollars will each of them receive? 82. Divide $87.50 among A, B, C, D, and E; and give B 1 dollar more than A, C 1 dollar more than B, D 1 dollar more than C, and E 1 dollar more than D. What number of dollars will each of them receive? 83. A merchant wishes to ship 750 bushels of flax-seed, in casks containing 7 bushels 2 pecks each; what number of casks will be required? 84. A farmer purchased 25 loads of manure, at $3.75 a load, and paid for the manure in apples at $1.75 a barrel; what was the number of barrels ? 85. How many pounds of butter, at 18 cents a pound, must be given in exchange for 2 barrels of flour, at $6.25 a barrel, and 75 pounds of sugar, at 8 cents a pound? Art. 265. 86. Boston is 71 degrees of longitude west from London; when it is 12 o'clock at Boston, what is the time at London? REMARK. Since the earth revolves on its axis from west to east once in 24 hours, must revolve one twenty-fourth part of its circumference in 1 hour, which is 15 degrees. Hence, if two places have a difference of longitude of 15 degrees, they must have a difference of time of 1 hour. If two places have a difference of longitude of 1 degree, they must have a difference of time of 4 minutes. If they differ in longitude only 1 minute of a degree, they must have a difference of time of 4 seconds. The time will be earlier at the place which is east of the other, because the earth revolves from west to east. 87. When it is 12 o'clock at Boston, what time is it at a place 45 degrees west of Boston? 88. Washington is 77 degrees west of London; what is their difference of time? When it is noon at London, what is the time at Washington? 89. A degree of longitude on the equator is equal to 60 geographical miles; what distance are the inhabitants who live on the equator carried every hour by the earth's revolution on its axis? 90. What number of miles are the inhabitants who live in the latitude of Boston carried every hour, by the revolution of the earth on its axis; the degree of longitude being equal to 44.42 miles? 91. How many yards of carpeting that is 1.5 yard wide, will be required to cover the floor of a room 18 feet in length, and 15 feet in width? 92. What number of yards of carpeting 1.25 yard in width, will be required to cover a floor 21 feet in length, and 17.5 feet in breadth? 93. What will be the expense of painting the walls of a room which measure 75.5 feet in compass, and 12.5 feet in height, at 10 cents a square yard? 94. The length of a room is 20 feet, its width 18 feet, its height 12 feet; what will the coloring of the walls come to, at 12 cents a square yard; deducting a fire-place 6 feet by 4 feet, and 4 windows, each 6 feet 6 inches by 3 feet 9 inches? 95. What will be the cost of paving a rectangular yard, the length being 21 yards, and the breadth 15 yards, in which there is a footpath 5 feet 3 inches wide running the whole length of the yard; the path to be paved with broad stones, at 36 cents a square yard; and the remainder of the yard with bricks, at 24 cents a square yard? 96. There is a block of marble 4 feet 6 inches in length, 2 feet 9 inches in breadth, and 2 feet 3 inches in depth; what will it cost, at 80 cents a cubic foot? 97. What will the bricks cost, at $6.25 a thousand, that will be sufficient to build a wall 120 feet long, 6 feet high, and 20 inches thick, allowing 27 bricks to build a cubic foot of the wall? 98. A merchant who owned of a ship, sold of his share for 2520 dollars. What was the value of the whole ship at the same rate? 99. A B and C, working together, can perform a piece of work in 12 days; A can perform the same work alone in 35 days, and B can perform the same work in 37 days. In what time can C perform the same work? 100. There is a ship which is valued at 8750 dollars, of which A owns .375. B purchases of A .125 of his share; what is the cost of B's share? 101. C purchased .625 of a ship, for which he paid 5450 dollars; what was the value of the whole ship? 102. A trader sold .375 of a yard of silk for .25 of a dollar; what is the value of a yard at that rate? 103. A ship of war having taken a prize, the captain received .0625 of the prize-money, which amounted to 500 dollars. What was the value of the whole prize? 104. A lady being asked her age, answered that she was 18 years old when she was married, and that she had lived with her husband of her whole life. What was the lady's age? 105. A spendthrift, having spent §, 4,, and of his money, had only 30 dollars left. How many dollars had he before spending any part of it? 106. A merchant deposited 5000 dollars in the Merchant's Bank. He has since drawn two checks upon the bank; the first for 12 per cent. of the 5000 dollars, and the second for 25 per cent. of what remained in the bank after the first check had been paid. What number of dollars has he remaining in the bank? 107. What must I pay annually for the insurance of of the value of my house against loss by fire, at a premium of 7 of 1 per cent.; the house being valued at $5750? 108. What is the duty on 4 hogsheads of brown sugar, weighing gross as follows: No. 1, 1224 pounds; No. 2, 1236 pounds; No. 3, 1272 pounds; and No. 4, 1296 pounds; draft 7 pounds per hogshead, tare 12 per cent., duty 24 cents a pound? 109. What is the duty on an invoice of woollen goods which cost in London £750. 15s. 6d. sterling, at 35 per cent. ad valorem? 110. A tax of of 1 per cent. has been assessed upon the whole amount of taxable property in a town; what must A pay, whose real estate is valued at $7650, and personal property at $1760? Boston, January 1, 1847. I promise to pay C D, or order, seven hundred and sixty dollars on demand, with interest at 6 per cent., for value received. 111. What will be due on the above note January 1, 1848? $5.25. $175.75. 112. If 12 bushels of salt be worth $8.625, what is the value of 327 bushels? 113. When a post 8 feet in height casts a shadow 12 feet in length, what must be the height of a steeple which casts a shadow 2361 feet, at the same time of day? 114. If 12.25 acres of land be worth $375.5, what is the value of a farm containing 175.75 acres? 115. If it require 124 pounds of wool to make a piece of flannel 267 yards in length, of a yard in width, what number of pounds of wool will be required to make 6 pieces of flannel, each piece measuring 273 yards, of a yard in width? 116. If 25 horses consume 153 tons of hay in 4 months, what number of tons will 37 horses consume in 3 months? 117. If 8 men can mow 72 acres of grass in 9 days, by working 9 hours each day, what number of men will be required to mow 96 acres of grass in 12 days, by working 12 hours each day? 118. A. B. purchased of C. D., on a credit of 6 months, the following bills of merchandise : Jan. 10, a bill amounting to $325.50. Feb. 15, a bill amounting to $450.75. April 10, a bill amounting to $225.50. June 15, a bill amounting to $175.25. On the 1st of July, A. B. wishes to give C. D. his note for the whole amount; at what time must it be made payable, so that no loss of interest shall be sustained by either party? 119. A trader purchased of an importing house the following bills of merchandise: May 1, a bill amounting to 150 dollars, on a credit of 180 days. June 15, a bill amounting to 175 dollars, on a credit of 120 days. July 10, a bill amounting to 125 dollars, on a credit of 90 days. Aug. 1, a bill amounting to 75 dollars, on a credit of 60 days. On the 15th, of September, the trader proposed to make a settlement, and give his note for the whole amount; at what time must the note be made payable? 120. A merchant purchased a quantity of merchandise for $725, and sold the same for $870 dollars; what per cent. did he gain? If he had sold the same for $580, what per cent. loss would he have sustained? 121. A merchant imported 860 yards of broadcloth, which cost £430 sterling, on which he paid a duty of 30 per cent.; the freight amounted to $17.20. He sold the same cloth at $4.50 a yard, on a credit of 6 months. What per cent. profit did he make, supposing he paid for the cloth 2 months before selling it, allowing money to be worth 6 per cent. a year in trade? 122. Sold a quantity of sugar for $84, and lost 12 per cent.; what should it have been sold for to have gained 12 per cent.? 123. A merchant sold flour at $5.50 a barrel, and gained 10 per cent.; he has since sold the same kind of flour at $6.25 a barrel. What per cent. did he gain at the latter price? 124. A man died leaving an estate of $20000 to be divided between his two sons, one being 11 and the other 16 years old, in such proportion that the share of each, being on interest at 6 per cent., should amount to the same sum, when they should arrive at 21 years of age; what sum must each receive? 125. A draper, by selling a piece of cloth for $36, lost of of 25 per cent. of what the cloth cost. What did he pay for the piece of cloth? |