710 The lateral area of a circular cylinder is equal to the product of the perimeter of a right section by an element. HYPOTHESIS. AK is a circular cylinder, S the lateral area, P the perimeter of a right section, and E an element. CONCLUSION. SPX E. PROOF Inscribe in the cylinder a prism whose base is a regular polygon, and let S' be its lateral area, and P' the perimeter of a right section. If the number of lateral faces of the prism is indefinitely increased, 711 COROLLARY 1. The lateral area of a cylinder of revolu tion is equal to the product of the circumference of its base by its altitude. 712 COROLLARY 2. Let S denote the lateral area, T the total area, H the altitude, and R the radius of a cylinder of revolution. Then S = 2 TR X H, and T2 R X H + 2 TR2 = 2 TR (H+ R). PROPOSITION XXXII. THEOREM 713 The volume of a circular cylinder is equal to the product of its base by its altitude. K HYPOTHESIS. AK is a circular cylinder, V the volume, B the base, and H the altitude. CONCLUSION. V=Bx H. PROOF Inscribe in the cylinder a prism whose base is a regular polygon, and let V' be its volume, and B' its base. Then V'B' × H. If the number of lateral faces of the prism is indefinitely B' approaches B as a limit. § 638 increased, § 455 ... B' × H approaches B x H as a limit. § 267 § 709 But V'B' × H. Proved 714 COROLLARY. Let V denote the volume, H the altitude, and R the radius of a cylinder of revolution. Then VR2 × H. 715 The lateral areas, or the total areas, of two similar cylinders of revolution are to each other as the squares of their altitudes, or as the squares of their radii; and their volumes are to each other as the cubes of their altitudes, or as the cubes of their radii. R R HI HYPOTHESIS. S and S' are the lateral areas, T and T' the total areas, V and V' the volumes, H and H' the altitudes, R and R' the radii of two similar cylinders of revolution. CONCLUSION. S: S'T: T′ = H2: H'2 = R2: R'2, PROBLEMS OF COMPUTATION 1228 Find the lateral area of a cylinder of revolution whose altitude is 12 inches, and the diameter of whose base is 4 inches. 1229 Find the total area of a cylinder of revolution whose altitude is 3 feet, and the diameter of whose base is 18 inches. 1230 Find the volume of a right circular cylinder whose altitude is 6 inches, and the diameter of whose base is 2 inches. 1231 A stand pipe is 12 feet in diameter and 48 feet high. How many gallons of water will it hold, estimating 71⁄2 gallons to a cubic foot? 1232 How many square feet of sheet iron in a smokestack 52 feet high and 6 feet in diameter ? 1233 A cylindrical cistern is 6 feet deep and 6 feet in diameter. How long will it take to fill it if 200 cubic inches of water flow into it per minute ? 1234 How many cubic yards of earth are removed in constructing a tunnel 135 yards long, a right section of which is a semicircle with a radius of 15 feet? 1235 A cylindrical oil tank is 28 feet long and 6 feet in diameter. How many gallons of oil does it hold, estimating 7 gallons to a cubic foot? 1236 Find the volume generated by a rectangle, whose dimensions are 4 and 7 inches respectively, in revolving (a) about its longer side as an axis; (b) about its shorter side as an axis. 1237 The altitudes of two similar cylinders of revolution are as 3: 5. What is the ratio of their volumes? of their total areas? 1238 The lateral area of a cylinder of revolution is 108 sq. ft., and its radius 3 ft. Find the lateral area of a similar cylinder whose radius is 5 ft. 1239 The altitudes of two similar cylinders of revolution are as 2:3. If the volume of the first is 16, find the volume of the second. 1240 The volumes of two similar cylinders of revolution are as 125 216. Find the ratio of their lateral areas. 1241 How much water will flow through a pipe in one minute, if it flows 2 feet per second, and the interior diameter of the pipe is 6 in. ? CONES DEFINITIONS 716 A conical surface is a surface generated by a moving straight line which constantly touches a fixed curve and passes through a fixed point not in the plane of the curve. The moving straight line is called the generatrix; the fixed curve, the directrix; and the fixed point, the vertex. Any straight line in the surface which corresponds to some position of the generatrix is called an element of the surface. When the generatrix is of indefinite length, the conical surface described consists of two portions called nappes, the upper nappe and the lower nappe. A Conical Surface It is customary, however, when we speak of a cone to refer to one nappe only. 717 A cone is a solid bounded by a conical surface and a plane cutting all its elements. The conical surface is called the lateral surface of the cone, and the plane surface is called the base of the cone. The vertex of a cone is the vertex of the conical surface. The altitude of a cone is the perpendicular drawn from the vertex to the plane of the base. 718 A circular cone is a cone whose base is a circle. A Cone The axis of a circular cone is the straight line which joins the vertex and the center of the base. The axis of a cone is not necessarily its altitude. |