RULE. To find the convex surface of a frustum, multiply the sum of the perimeters or circumferences of the bases by one-half the slant height. WRITTEN EXERCISES. 1. Find the convex surface of the frustum of a square pyramid whose slant height is 24 ft., and the side of the lower base 10 ft. and upper base 6 ft. 2. Find the convex surface of the frustum of a cone whose slant height is 42 ft., and the radii of the bases 12 and 4 ft., respectively. 3. Find the entire surface of the frustum of a cone whose slant height is 50 ft., and the radii of the bases 10 and 5 ft., respectively. 4. Find the entire surface of the frustum of a triangular pyramid whose slant height is 40 in., and the sides of the upper base 4 in. and of the lower base 10 in. RULE. 263. To find the volume of a frustum, take the sum of the areas of the two bases, to which add the square root of their product, and multiply this sum by one-third of the altitude. WRITTEN EXERCISES. 1. Find the volume of the frustum of a square pyramid the sides of whose bases are 3 ft. and 6 ft. and altitude 12 ft. The volume = = = 2. Find the contents of the frustum of a cone the radii of whose bases are 3 and 4 ft. and altitude 18 ft. The contents 222 π (32 + 62 + √32 × 62) 12 = (9 + 36 + 18) 4 · = 252 cu.ft. = Π (π 32 + π 42 + Vπ2 32 × 42) 1⁄8 = (9 π + 16 π + 12 π) 6 697.4352 cu. ft. 3. Find the contents of the frustum of a square pyramid the sides of whose bases are 2 and 8 ft. and altitude 15 ft. 4. Find the contents of a log the radii of whose bases are 2 and 3 ft. and length 60 ft. 5. How many cubic feet in the frustum of a cone the radii of whose bases are 12 and 8 ft. and altitude 27 ft.? 6. Find the contents of the frustum of a triangular pyramid the sides of whose bases are 4 and 6 ft. and altitude 21 ft. RULE. The surface of a sphere is equal to the square of the diameter, or four times the square of the radius, multiplied by 3.1416. WRITTEN EXERCISES. 1. Find the surface of a sphere whose radius is 5 in. 2. Find the surface of a sphere whose radius is 12 in. 3. Assuming the earth to be a sphere 7960 miles in diameter, how many square miles in its surface? 4. What will it cost to plate with gold a sphere 18 in. in diameter at $40.50 a square foot? 5. What will it cost to tin the hemispherical dome of an observatory 40 ft. in diameter, at 10 cents a square foot? RULE. 265. The volume of a sphere is equal to one-sixth the cube of its diameter, multiplied by 3.1416. WRITTEN EXERCISES. 1. Find the volume of a sphere whose radius is 10 in. 2. The outer diameter of a spherical shell is 12 in. and the inner diameter is 8 in.; find the contents. 3. Find the weight of a ball of gold 8 in. in diameter, if a cubic foot weighs 1204 lb. 4. Find the weight of a cannon-ball of cast iron 15 in. in diameter, if a cubic foot weighs 450 lb. It is readily seen that then WRITTEN EXERCISES. 1. Find the side of the largest cube that can be cut from a sphere 15 in. in diameter. A B2+ BC2 + CD2 3 A B A 1 AD2. But A B = BC = CD; 15= 225, A B2 = 75, AB V75 - 8.66+ in. 2. Find the side of the largest cube that can be cut from a sphere 30 in. in diameter. PROBLEMS IN MENSURATION. 1. How many acres in a triangle whose base is 40 chains and altitude 36 chains? 2. How many square yards in a triangle whose sides are each 50 feet? 3. How many acres in a triangle whose sides are respectively 20, 30, and 40 chains? 4. The hypotenuse of a right triangle is 312 feet and the base is 282 feet; what is the perpendicular? 5. Find the diagonal of a square whose side is unity. 6. Find the diagonal of a cube whose side is unity. 7. How many acres in a parallelogram whose base is 161.5 rods and altitude 106.3 rods? 8. The area of a rectangle is 1872 square rods, and the length is to the width as 4 is to 3; required the length of the sides. 9. The area of a circle is 2 A. 3 R. 15 P.; what is the circumference? 10. Find the side of a square whose area is equal to the area of a circle 40 feet in diameter. 11. Find the length of an arc of 35° in a circle whose radius is 40 rods. 12. Find the area of the largest possible square that can be cut from a circle 62 feet in circumference. 13. Find the diameter of a circle whose area is numerically equal to 10 times its circumference. 14. How many square feet in the bottom and convex surface of a scrap-basket whose lower base is 9 inches in diameter, upper base 12 inches in diameter, and altitude 10 inches? 15. A piece of timber 90 feet long is 6 by 8 inches at the smaller end and 18 by 24 inches at the larger end; how many cubic feet does it contain? 16. Find the diameter of a sphere whose surface and volume are numerically equal. 17. Find the diagonal of a parallelopiped whose base is 5 feet by 7 feet, and altitude 16 feet. 18. Required the entire surface of a square pyramid whose altitude is 36 inches and side of the base 30 inches. 19. Required the dimensions of a cubical bin which will hold 1000 bushels of grain. 20. Find the contents of a cone whose altitude is 60 feet and the radius of the base 20 feet. 21. Find the volume of the frustum of a cone whose altitude is 24 feet, the radius of the lower base being 8 feet and of the upper base 6 feet. 22. If a conical hay-stack 24 feet in diameter and 20 feet high contains 12 tons, what are the dimensions of a similar stack which contains 20 tons? 23. A circular mirror 24 inches in diameter is surrounded by a frame 3 inches wide; what is the cost of the frame, at $2 a square foot? 24. A garden containing 74 square feet less than one-fourth of an acre is planted with hyacinths in squares 6 inches apart; how many plants will be required, if the outer rows are 6 inches from the edge? 25. The largest possible square piece of timber of uniform size is cut from a log 40 feet long, 15 inches in diameter at one end, and 18 inches at the other; how much is wasted? 26. A cylinder is 6 feet in diameter and 8 feet high; what are the dimensions of a similar cylinder whose surface is 64 times as much? 27. How much material will be required to make a hollow sphere whose outside diameter is 15 inches and thickness 1 inch? 28. What is the value of a gold sphere 4 inches in diameter, if a sphere of silver 1 inch in diameter is worth $5, and the value of gold is to the value of silver as 16 to 1? |