Ex. 1646. The area AA'B'B, bounded by two concentric arcs and parts of two radii is folded so as to form the lateral surface of a frustum of a cone whose lower base equals 36 π. Find 20, and the volume of the frustum if OA = 5 and OA' = 10. Ex. 1647. By folding a piece of paper similar to the figure of the preceding exercise we wish to obtain a frustum of a cone, whose slant height equals 5, and whose bases have the radii 9 and 6 respectively. Find OA, OB, and ≤ 0. Ex. 1648. A cylindrical tin can closed at the top requires the least amount of tin if its height equals its diameter. Find the diameter of such a can which holds one quart. (1 qt. 57.75 cu. in.) = Ex. 1649. How many square feet of ground are covered by a conical tent 8 ft. high, if the canvas contains 1884 sq. ft.? (Assume π = 34.) Ex. 1650. A conical vessel whose height equals the diameter of the base contains water to a depth of 6 in. Upon immersing a cube the water rises 2 in. Find the edge of the cube. Ex. 1651. A cylindrical boiler which is partly filled with water has a length of 6 ft. and a diameter of 2 ft. Find the volume of the water if its greatest depth is 6 in. and the axis of the cylinder has a horizontal position. Ex. 1652. A cylindrical log of wood, 10 ft. long and 1 ft. in diameter, floats on water. Find the volume and the weight of the displaced water if 9 in. of the vertical diameter are immersed in water. (1 cu. ft. of water weighs 62.5 lb.) BOOK VIII THE SPHERE 693. DEF. A sphere is a solid bounded by a surface, all the points of which are equally distant from a point. The fixed point is called the center of the sphere. 694. DEF. The radius of a sphere is a straight line drawn from the center to any point in the surface. 695. DEF. The diameter of a sphere is a straight line passing through the center and terminated at either end by the surface. 696. From the definitions it follows that (1) All the radii of a sphere are equal, and all diameters are equal. (2) A semicircle rotating about its diameter generates a sphere. (3) Two spheres are equal if their radii or their diameters are equal, and conversely. (4) A point is without a sphere if its distance from the center is greater than the radius. Ex. 1653. The radii of two spheres are respectively 10 in. and 4 in., their line of centers (i.e. the line joining their centers) is 7 in. Does every point of the smaller sphere lie within the larger one? PROPOSITION I. THEOREM 697. Every section of a sphere made by a plane is Given CBD, the intersection of plane MN, and a sphere whose center is 0. Since any two points B and C in CBD are equidistant from 4, all points must be equidistant from 4. Or CBD is a circle. Q. E. D. 698. COR. 1. Circles of a sphere made by planes equidistant from the center are equal; and of two circles made by planes not equidistant from the center the circle made by the plane nearer to the center is the greater. For since 400-A02, AC is the smaller, the greater AO. = 699. DEF. A great circle of a sphere is a section made by a plane passing through the center. 700. DEF. A small circle of a sphere is a section made by a plane not passing through the center. 701. DEF. The axis of a circle of a sphere is the diameter perpendicular to the plane of the circle; its ends are the poles of the circle. 702. COR. 2. The axis of a circle passes through the center of the circle. 703. COR. 3. 704. COR. 4. other. All great circles of a sphere are equal. Any two great circles of a sphere bisect each For since the plane of each contains the center of the sphere, their intersection is a diameter and bisects both circles. 705. COR. 5. Every great circle bisects the sphere. 706. COR. 6. One and only one circle may be drawn through any three points in the surface of a sphere. (A plane is determined by three points.) 707. COR. 7. A great circle may be drawn through two points B and C in the surface of a sphere. (Three points, B, C, and the center O, determine a plane.) Generally there is only one great circle which passes through two given points, but if the given points are ends of a diameter, any number of great circles can be passed through these points. 708. DEF. The distance between two points on the surface of a sphere is the length of the minor arc of a great circle between them. Ex. 1654. Considering the earth as a sphere, what kind of circles are the meridians? the parallels of latitude? the equator? What are the poles of the parallels of latitude ? Ex. 1655. What is the radius of a small circle, if the distance of its plane from the center of the sphere is 9 in., and the radius of the sphere is 15 in.? Ex.1656. What is the area of a section of a sphere made by a plane, if the distance of the plane from the center of the sphere is 3, and the radius of the sphere is 4? PROPOSITION II. THEOREM 709. All points in the circumference of a circle of a sphere are equidistant from a pole of the circle. Given P and P', the poles of circle ABC of a sphere. To prove and arc PA arc PB =arc PC, arc P'A arc P'Barc P'C. HINT. Prove by means of (509) the equality of straight lines PA, PB, and PC. 710. DEF. The polar distance of a circle of a sphere is the distance of a point in the circumference from the nearer pole. 711. REMARK. A quadrant in Spherical Geometry is the fourth part of the circumference of a great circle. 712. COR. The polar distance of a great circle is a quadrant. Ex. 1657. The polar distance of a circle of a sphere is 60°, and the radius of the sphere is 13 in. Find (a) the distance of its plane from the center. (b) the radius of the circle. |