701. The lateral areas, or the total areas, of 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. Let S, S' denote the lateral areas, T, T' the total areas, V, V' the volumes, H, H' the altitudes, R, R' the radii, of two similar cylinders of revolution. Proof. Since the generating rectangles are similar, we have by $$ 351, 335, PROBLEMS OF COMPUTATION. Ex. 678. The diameter of a well is 6 feet, and the water is 7 feet deep. How many gallons of water are there in the well, reckoning 74 gallons to the cubic foot? Ex. 679. When a body is placed under water in a right circular cylinder 60 centimeters in diameter, the level of the water rises 40 centimeters. Find the volume of the body. Ex. 680. How many cubic yards of earth must be removed in constructing a tunnel 100 yards long, whose section is a semicircle with a radius of 18 feet? Ex. 681. How many square feet of sheet iron are required to make a funnel 18 inches in diameter and 40 feet long? Ex. 682. Find the radius of a cylindrical pail 14 inches high that will hold exactly 2 cubic feet. Ex. 683. The height of a cylindrical vessel that will hold 20 liters is equal to the diameter. Find the altitude and the radius. Ex. 684. If the total surface of a right circular cylinder is T, and the radius of the base is R, find the height of the cylinder. Ex. 685. If the lateral surface of a right circular cylinder is S, and the volume is V, find the radius of the base and the height. Ex. 686. If the circumference of the base of a right circular cylinder is C, and the height H, find the volume V. Ex. 687. Having given the total surface T of a right circular cylinder, in which the height is equal to the diameter of the base, find the volume V. Ex. 688. If the circumference of the base of a right circular cylinder is C, and the total surface is T, find the volume V. Ex. 689. If the volume of a right circular cylinder is V, and the altitude is H, find the total surface T. Ex. 690. If V is the volume of a right circular cylinder in which the altitude equals the diameter, find the altitude H, and the total surface T. Ex. 691. If T is the total surface, and H the altitude of a right circular cylinder, find the radius R, and the volume V. L CONES. 702. DEF. A conical surface is the 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 which generates the conical surface is called the generatrix, the fixed curve the directrix, and the fixed point the vertex. 703. DEF. The generatrix in any position is called an element of the conical surface. If the generatrix is of indefinite length, the surface consists of two portions, one above and the other below the vertex, which are called the upper and lower nappes, respectively. Conical Surface. 704. DEF. If the directrix is a closed curve, the solid bounded by the conical surface and a plane cutting all its Cones. elements is called a cone. 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 the conical surface is called the vertex of the cone, and the elements of the conical surface are called the elements of the cone. The perpendicular distance from the vertex to the plane 705. DEF. A circular cone is a cone whose base is a circle. The straight line joining the vertex and the centre of the base is called the axis of the cone. If the axis is perpendicular to the base, the cone is called a right cone. If the axis is oblique to the base, the cone is called an oblique cone. Simila Cones of Revolution. 706. DEF. A right circular cone is a cone whose base is a circle and whose axis is perpendicular to its base. A right circular cone is called a cone of revolution, because it may be generated by the revolution of a right triangle about one of its legs as an axis. The hypotenuse of the revolving triangle in any position is an element of the surface of the cone, and is called the slant height of the cone. The elements of a cone of revolution are all equal. 707. DEF. Similar cones of revolution are cones generated by the revolution of similar right triangles about homologous legs. Tangent Plane. 708. DEF. A tangent line to a cone is a straight line, not an element, which touches the lateral surface of the cone but does not cut it. 709. DEF. A tangent plane to a cone is a plane which contains an element of the cone but does not cut the surface. The element contained by the plane is called the element of contact. 710. DEF. A pyramid is inscribed in a cone when its lateral edges are elements of the cone and its base is inscribed in the base of the cone. 711. DEF. A pyramid is circumscribed about a cone when its base is circumscribed about the base of the cone and its vertex coincides with the vertex of the cone. 712. DEF. A truncated cone is the portion of a cone included between the base and a plane cutting all the elements. A frustum of a cone is the portion of a cone included between the base and a plane parallel to the base. Circumscribed Pyramid. Inscribed Pyramid. 713. DEF. The base of the cone is called the lower base of the frustum, and the parallel section is called the upper base of the frustum. 714. DEF. The altitude of a frustum of a cone is the perpendicular distance between the planes of its bases. 715. DEF. The lateral surface of a frustum of a cone is the portion of the lateral surface of the cone included between the bases of the frustum. 716. DEF. The elements of a cone between the bases of a frustum of a cone of revolution are equal, and any one is called the slant height of the frustum. A plane which cuts from the cone a frustum cuts from the inscribed or circumscribed pyramid a frustum. |