EXAMPLES. 1. What is the volume of a pyramid whose altitude is 16 in. and the area of the base 25 sq. in. ? 2. What is the volume of a cone whose altitude is 24 ft. and the diameter of the base 8 ft. ? 3. What is the volume of a rectangular pyramid whose altitude is 20 ft. and the sides of the base 4 ft. and 5 ft., respectively? 4. Find the volume of a cone whose altitude is 50 ft. and the radius of the base 7 ft. 5. Find the volume of a pyramid whose base is a triangle, each side of which is 6 ft. and whose altitude is 15 ft. 485. To find the Volume of a Frustum of a Pyramid or Cone. It can be shown by geometry that the frustum of a pyramid or cone is equal to the sum of three pyramids or cones whose common altitude is the altitude of the frustum, and whose bases are the lower base, the upper base, and a mean proportional between the bases of the frustum. Hence, RULE.-To the sum of the areas of the two ends add the square root of their product, and multiply this sum by one-third of the altitude. EXAMPLES. 1. What is the solid contents of the frustum of a square pyramid the sides of whose bases are 5 ft. and 8 ft., and whose altitude is 21 ft.? 2 Find the volume of the frustum of a square pyramid the side of whose bases are 12 ft. and 18 ft., and altitude 30 ft. 3. Find the number of cubic feet in a log 15 ft. long, the diameter of the larger end being 3 ft., and of the smaller end 2 ft. 486. To find the Volume of a Sphere. A sphere may be regarded as composed of an infinite number of pyramids whose bases form the surface of the sphere and whose altitude is the radius of the sphere. Hence, RULE.-Multiply the convex surface by one-third of the radius; or, multiply the cube of the diameter by one-sixth of 3.1416. EXAMPLES. 1. What is the volume of a sphere whose diameter is 8 inches? 2. The circumference of a sphere is 20 inches. Find its cubical contents. 3. The outer diameter of a spherical shell is 10 in. and the inner diameter is 5 in. What are the cubical contents of the shell? 4. If a cubic foot of cast iron weighs 450 lb., what is the weight of a cannon-ball whose diameter is 12 inches? 487. To find the Side of a Cube inscribed in a given Sphere. A cube is inscribed in a sphere when the vertices of its angles are in the surface of the sphere. It is seen by Art. 438 that AB + BC2 + CD2 2 BC: = CD. Therefore, Hence, 3 2 113 RULE.-Divide the square of the diameter of the sphere by 3, and extract the square root. EXAMPLES. 1. What is the side of the largest cube that can be cut from a sphere 9 in. in diameter? 2. Find the side of the largest cube that can be cut from a wooden ball 40 in. in circumference. 488. To find the Volume of an Irregular Body. RULE.-1. Place the body in a suitable vessel, pour in water until the body is just covered, and find the space occupied by both body and water. 2. Remove the body from the vessel and find the space occupied by the water alone. The difference will be the volume of the body. 1. A vessel in the form of a cylinder is most convenient for finding the volume of irregular bodies. 2. The volume may often be most conveniently found by taking the diameter of the vessel and the difference between the two altitudes of the water. Art. 474 This will give the volume of the body at once. 3. A body lighter than water must be held down by pressure at the highest poinu, EXAMPLES. 1. An irregularly shaped stone was placed in a cylindrical vessel 20 inches in diameter, and covered with water. The height of the water was then 15 inches; but when the stone was removed, the height of the water was only 8 inches. What was the volume of the stone? 2. A block of marble was placed in a 10-gallon vessel. It then required 4 gallons of water to fill the vessel. What was the volume of the block of marble? 489. To find the Quantity of Wood. 490. A Cord of wood is a pile 8 feet long, 4 feet wide, and 4 feet high. It contains 8 cord feet, or 128 cubic feet. 491. A Cord Foot is a cross section of this pile 1 foot long, and contains 16 cubic feet. 8 FT. LONG ONE CORD EXAMPLES 492. 1. How many cords of wood in a pile 20 ft. long, 4 ft. wide, and 6 ft. high? 2. A certain pile of wood is 36 ft. long, 8 ft. wide, and 6 ft. high. What is its value at $3.50 per cord? 3. How many cords of wood can be placed in a shed 28 ft. long, 20 ft. wide, and 14 ft. high? 4. A man sold 10 loads of wood. Each load was 8 ft. long, 4ft. wide, and 5 ft. high. What was the value of the wood at $4.25 a cord? 5. A tanner filled a shed 100 ft. long, 60 ft. wide, and 18 ft. high with bark which cost him $5.25 a cord. How much did the bark cost him? BOARDS AND TIMBER. 493. Boards, Planks, Scantling, Joists, and other sawed timber are estimated by what is called Board Measure, the unit of which is the Board Foot. 494. A Board Foot is 1 ft. long, 1 ft. wide, and 1 in. thick. A board an inch thick is taken as the standard, and its contents in board feet are the product of its length and breadth in feet, or the number of square feet it contains. The same measurement applies when the board is less than 1 inch thick. Thus a board 16 ft. long, 14 in. wide, and 1 in. or less in thickness contains 161183 feet board measure, or 183 sq. ft. When lumber is more than one inch thick the thickness is taken into account. = Thus, a board 16 ft. long, 14 in. wide, and 11⁄2 in. thick contains 16 × × = 28 ft. board measure. A cubic foot of lumber contains 12 board feet. Hence, board feet may be reduced to cubic feet by dividing by 12, and cubic feet to board feet by multiplying by 12. 495. To find the number of Board Feet in a Board. RULE.-Multiply the length in feet by the width in inches, and divide by 12. When a board tapers evenly, the mean or average width is used, which is half the sum of the two ends. EXAMPLES. Find the number of feet in the following boards: 7. How many feet in a board 16 ft. long, 15 in. wide, and in. thick ? 8. How many feet in a board 18 ft. long, 16 in. wide at one end and tapering to 10 in. at the other? 9. Find the cost of 19 boards, each 14 ft. long, 15 in. wide, at $2.30 per C. 10. How much will it cost to lay 3 floors, each 20 ft. long and 16 ft. wide, at $25 per M, allowing for grooving? 496. To find the number of Board Feet in Timbers. RULE.-Multiply the length in feet by the width and thickness in inches and divide by 12. EXAMPLES. 1. How many board feet in a plank 14 feet long, 8 in. wide, and 2 in. thick ? 2. How many board feet in a joist 20 ft. long, 9 in. wide, and 3 in. thick ? 3. Find the number of board feet in a joist 24 ft. long, 7 in. wide, and 4 in. thick ? 4. How many board feet in a piece of timber 18 ft. long, and 10 in. square? 5. What will be the cost of 10 girders, each 40 ft. long, 14 in. wide, and 12 in. thick, at $30 per M? 497. To find the Cubical Contents of Square Timber. RULE.-Multiply the area of one end in inches by the length in feet, and divide by 144. EXAMPLES. 1. What are the cubical contents of a stick of square timber 18 ft. long and 16 in. square? 2. What are the cubical contents of a stick of timber 20 ft. long and 14 in. square? 3. Find the cubical contents of a piece of timber, 12 by 14, and 24 feet in length. 4. Find the cubical contents of 3 pieces of timber, each 9 by 12, and 16 ft. long. ROUND TIMBER. 498. The contents of "Saw Logs" are usually estimated by the quantity of square-edged inch boards they will produce. There is no uniform nor accurate rule for determining the contents of a log Board Measure. Lumbermen generally use "Log Books" containing prepared tables showing the number of feet board measure which logs of various dimensions will produce; but they are not accurate. Too much is allowed for waste. 499. To find the number of Board Feet in a Log. The following rule is often used; but like others is not strictly accurate, though perhaps one of the best. RULE.-Multiply the square of two-thirds of the smaller diameter, in inches, by the length in feet and divide by 12. 1. On large logs this allows too much for waste. Some dealers, on this account, deduct only 4 inches from the diameter, instead of one-third. |