EXERCISES. THEOREMS. 1. The lateral area of a cylinder of revolution is equal to the area of a circle whose radius is a mean proportional between the altitude of the cylinder and the diameter of its base. 2. If the slant height of a cone of revolution is equal to the diameter of its base, its lateral area is double the area of its base. 3. The volume of a cylinder of revolution is equal to the product of its lateral area by half its radius. 4. A plane through two elements of a cylinder of revolution cuts the base in a chord which subtends at its centre π an angle of compare the lateral areas of the two parts 3 of the cylinder. 5. A rectangle revolves successively about two adjacent sides whose lengths are a and b compare the volumes of the two cylinders that are generated. 6. The two legs of a right triangle are a and b find the area of the surface generated when the triangle revolves about its hypotenuse. 7. Prove that a sphere may be inscribed in a cylinder of revolution, and that it will touch it along the circumference of a great circle. 8. The lateral area of a given cone of revolution is double the area of its base: find the ratio of its altitude to the radius of its base. 9. On each base of a frustum of a cone of revolution, a cone stands having its vertex in the centre of the other base find the radius of the circle of intersection of the two cones, the radii of the bases being r and r2 : 10. If the altitude of a cylinder of revolution be equal to the diameter of its base,* the volume is equal to the product of its total area by one-third of its radius. 11. If the slant height of a cone of revolution be equal to the diameter of its base, its total area is to the area of the inscribed sphere as 9: 4. 12. In a frustum of a cone of revolution the inclination of the slant height to one base is 45°: find the lateral area, the radii of the bases being r, and r. 13. If the radius of a sphere is bisected at right angles by a plane, the two zones into which the surface of the sphere is divided are to each other as 3 : 1. 14. If a cylinder and cone, each equilateral, be inscribed in a sphere, the total area of the cylinder is a mean proportional between the total area of the cone and the area of the sphere. The same is true of the volumes of these bodies. 15. If a cylinder and cone, each equilateral, be circumscribed about a sphere, the total area of the cylinder is a mean proportional between the total area of the cone and the area of the sphere. The same is true of the volumes. 16. A cone of revolution whose vertical angle is 60°, is circumscribed about a sphere: compare the area of the sphere and the lateral area of the cone. Compare their volumes. 17. The base of a cone is equal to a great circle of a sphere, and the altitude of the cone is equal to a diameter of the sphere: compare the volumes of the cone and sphere. 18. The volume of a cone of revolution is equal to the area of its generating rectangle multiplied by the circumference generated by the point of intersection of the diagonals of the rectangle. 19. The volume of a sphere is to the volume of the cir cumscribed cube as π: 6. *Called an equilateral cylinder. + An equilateral cone, 20. The volume of a sphere is to the volume of the inscribed cube as π: 2. 21. If d is the distance of a point P from the centre of a sphere whose radius is R, the sum of the squares of the six segments of three chords at right angles to each other passing through P is 6R-2ď2. 22. If h is the height of an aeronaut, and R is the radius of the earth, the extent of surface visible NUMERICAL EXERCISES. = R+h 23. Find the lateral area, total area, and volume, of a cylinder of revolution, the radius of the base being 4 and the altitude 10. 24. Find the lateral area, total area, and volume, of a cone of revolution, the radius of the base being 4 and the altitude 10. 25. Find the lateral area, total area, and volume, of a frustum of a cone of revolution, the radii of the bases being 7 and 2 and the altitude 3. 26. Find the lateral area of a frustum of a cone of revolution, the radii of the bases being 21 and 6 inches and the altitude 36 inches. Ans. 3308.1 cu. ins. 27. Find the volume of a frustum of a cone of revolution, the radii of the bases being 4 and 2 feet and the altitude 9 feet. Ans. 65.97 cu. ft. 28. The slant height of a cone of revolution is 4 feet: how far from the vertex must the slant height be cut by a plane parallel to the base that the lateral area may be divided into two equivalent parts? 29. The altitude of a cone of revolution is equal to the diameter of its base: find the ratio of the area of the base to the lateral area. 30. Find the volume of an equilateral cylinder in terms of its total area. See Ex. (10). 31. The lateral area of a cylinder of revolution is 116 sq. ft., and the altitude is 14 feet: find the diameter of its base. Ans. 2.65 ft. 32. The volume of a cylinder is 15.7 cu. ft., and the diameter of its base is 2 feet: find its altitude. Ans. 5 feet. 33. The lateral area of a cylinder of revolution is 707 and its volume is 1757: find its height and the radius of its base. 34. The lateral area of a cone of revolution is 607 and its slant height is 12: find its volume. 35. Find the total area of a frustum of a cone of revolution, the radii of its bases being 9 and 4 feet and its height 12 feet. Ans. 266π. 36. The volume of a frustum of a cone of revolution is 920 cu. ft., and its height is 12 feet: find the radii of its bases if their sum is 8 feet. 37. Find the number of cubic feet in a log 12 feet long and 6 feet in diameter. Ans. 418.88 cu. ft. 38. Find the number of cubic feet in the trunk of a tree, 70 feet long, the diameters of its ends being 10 and 7 feet. 39. How many square inches of sheet-iron does it take to make a joint of 6-inch stovepipe 2 feet long, allowing an inch and a half for the seam? 40. The heights of two cylinders of revolution of equal volumes are as 9:16; the diameter of one of them is 6 feet: find the diameter of the other. 41. A cylinder of revolution whose base is 5 feet in diameter and a cone of revolution whose base is 6 feet in diameter, have equal volumes: the height of the cone is 10 feet. find the height of the cylinder. 42. The height of a frustum of a cone of revolution is 6 feet, and the diameters of its bases are 3 and 2 feet: find the height of a cylinder of revolution of the same volume as the frustum, and whose base is equal to the mid-section of the frustum. See (777). 43. The volumes of two equilateral cylinders are to each other as 3: 4 find the ratio of their heights. 44. The volumes of two similar cones are 27 cubic feet and 216 cubic feet, and the height of the first is 9 feet: find the height of the other. 45. The volumes of two similar cones of revolution are to each other as 512: 729 find the ratio of their lateral areas. 46. The slant heights of two similar cones of revolution are to each other as 3:5: find the ratio of their lateral areas, and of their volumes. 47. The height of a frustum of a cone is the height of the complete cone: find the ratio of the volume of the frustum to that of the cone. 48. The total areas of two similar cylinders of revolution are to each other as 25: 49: find the ratio of their volumes. 49. The altitude of a cone of revolution is 10, and its slant height is 14: find the total area of the inscribed cylinder whose altitude is 6. 50. The volume of a cone of revolution is 392, and its slant height is to the diameter of its base as 100:56: find its altitude and the diameter of its base. 51. Find the surface and volume of a sphere whose di- * ameter is (1) 16 inches; and (2) 17 inches. Ans. (1) 8041; 2144: (2) 908; 2572.45. 52. Find the diameter of a sphere if the surface is (1) 1809 square inches; (2) 616 square inches; and (3) 9856 square inches. 53. The volume of a sphere is 113: find its diameter and its surface. 54. The volume of a sphere is 7767: find its diameter and its surface. 55. The surface of a sphere is 7847: find its radius and its volume. |