The volumes of solid figures can be found by applying the following rules. Rules. - 1. To find the volume of a prism or cylinder, multiply the area of the base by the altitude. 2. To find the volume of a pyramid or cone, multiply the area of the base by one third the altitude. 3. To find the volume of the frustum of a pyramid or cone, add to the sum of the areas of the bases the square root of their product, and multiply the sum by one third the altitude of the frustum. 4. To find the volume of a sphere, multiply the cube of the diameter by .5236. FRUSTUM OF A CONE 5. To find the volume of a spherical segment, add the square of its altitude to three times the square of the radius of the base of the segment; multiply this sum by the altitude, and the product by .5236. 6. To find the volume of a torus or cylindrical ring, multiply the area of the cross section by half the sum of the inner and outer circumferences. Exercise 1. What is the volume of a cube 14 in. on an edge? 2. What is the volume of a block 3 ft. by 5 ft. by 6 ft.? TRIANGULAR PYRAMID 4. How many cubic yards of earth in a roadway cut 14 ft. wide, 42 ft. long, and 8 ft. 6 in. deep? 5. How many pounds of air in a room 24 ft. by 32 ft. by 14 ft. if 12 cu. ft. of air weigh 1 lb. ? Find the total areas and volumes of the following prisms: 6. Base 12 in. square, height 18 in. 7. Base 141⁄2 in. square, height 7 in. 8. Base a triangle with sides 5 in., 12 in., 13 in., height 26 in. 9. Base an equilateral triangle, sides 5 ft., height 22 ft. 10. Base a regular hexagon, sides 10 in., height 10 ft. 11. Find the weight of a castiron hexagonal prism of the dimensions given in the preceding example, if cast iron weighs 450 lb. per cubic foot. 12. Find the weight of a regular octagonal prism of granite, sides of the base being 6 in., and height 4 ft. 6 in., if granite weighs 170 lb. per cubic foot. CIRCULAR CONE 13. Find the volume of a cylinder 10 in. in diameter and 12 in. high. 14. What is the weight of a cylindrical barrel of tar 2 ft. in diameter and 2 ft. 6 in. in height, if tar weighs 65 lb. per cubic foot? 15. What is the weight of a granite pillar 2 ft. in diameter and 10 ft. high? (See Ex. 12.) 16. What is the capacity in gallons of a pipe-line 6 in. in diameter and 3 mi. long? 17. Find the weight of a mile of aluminum wire in. in diameter, if aluminum weighs 165 lb. per cubic foot. HEXAGONAL PRISM 18. A hollow cast iron cylinder open at both ends is 6 ft. long and 4 ft. in diameter, outside measurements. What will it weigh if it is 1 in. thick? (See Ex. 11.) 19. How many pyramids of ice cream can be made from 5 gallons of material if the pyramids have square bases 21 in. on a side, and an altitude SPHERICAL SEGMENT 21. If the conical crater of a volcano is 2 mi. in diameter and mi. deep, how many tenths of a cubic mile of material will it take to fill it? 22. A quantity of fine coal is thrown up against the side of a bin. The highest point of the coal is 6 ft., where it touches HOLLOW CYLINDER the side of the bin, and the slant height of the heap is 9 ft. If the base of the heap is semicircular in shape, how many tons does the pile contain if a ton of coal occupies 36 cu. ft.? 23. Find the volume of a pyramid whose base and faces are all equilateral triangles with sides 10 in. long. 24. Find the volume of the frustum of a square pyramid if the sides of its bases are 40 ft. and 16 ft., and the altitude of a lateral face is 20 ft. 25. A bucket is 16 in. deep, 18 in. in diameter at the top, and 12 in. in diameter at the bottom. How many gallons will it hold? 26. Find the weight of a pine flagstaff 120 ft. long, 24 in. in diameter at the base and 8 in. at the top, if pine weighs 35 lb. per cubic foot. 27. Determine the capacity in gallons of a pail 14 in. in diameter at the top, 10 in. in diameter at the bottom, 10 in. high. What is the capacity of a pail similar in shape but with all the dimensions half as great? QUADRANGULAR PRISM 28. How many cubic yards of earth will be required to fill 10,000 flower-pots the inside diameter of whose upper and lower bases are 10 in. and 6 in. and whose depth is 10 in. ? 29. Find the volume of a sphere 14 in. in diameter. 30. What is the weight of a cast brass sphere 6 in. in diameter if a cubic foot of brass weighs 500 lb. ? 31. What is the weight of an ivory sphere 3 in. in diameter, if ivory weighs 114 lb. per cubic foot? 32. Find the weight of a cast-iron dumb-bell if the bells are 5-inch spheres connected by a cylindrical bar 51 in. long and 13 in. in diameter. (See Ex. 11.) 33. What is the weight of 10,000 marbles in. in TORUS OR CYLINDRICAL RING diameter if the material of which the marbles are made weighs 170 lb. per cubic foot? 34. At the rate of 250 cu. ft. per minute how long will it take to inflate a spherical balloon 51 ft. in diameter ? |