EXERCISES. NUMERICAL. 842. Find the lateral area and the total area of a cylinder of revolution whose altitude is 18 in. and the diameter of its bases 12 in. 843. What is the volume of the same cylinder ? 846. What should are the radii of two similar of revolution whose altitude is 10 ii altitudes. V and the 500 sq. in.? 847. The altitude of a cylinder of revolution is three times its diameter; the total area is 1200 sq. in. Find the altitude and diameter. 848. The altitudes of two similar cylinders of revolution are as 7 to 5. What is the ratio of their total areas ? Of their volumes? 849. What formula expresses the total area of a cylinder of revolution whose altitude and radius are equal? 850. What formula expresses the volume of the same cylinder? 851. What is the ratio of the volume of the same cylinder to the volume of a cube having the same altitude? 852. Find the lateral area and the total area of a cone of revolution whose altitude is 15 in., and the diameter of whose base is 12 in. 853. Find the volume of the same cone. 854. The total area of a cone of revolution is 400 sq. in.; its altitude is 10 in. What is the diameter of its base? 855. What should be the altitude of a cone of revolution whose base has a diameter of 10 in., so that the lateral area may be a square foot? 856. What should be the radius of the base of a cone of revolution whose altitude is 10 in., so that its total area shall be 100 sq. in. ? 857. The altitude of a cone of revolution is four times the radius of its base; the lateral area is 500 sq. in. Find the radius and altitude. 858. The altitudes of two similar cones of revolution are as 11 to 8. What is the ratio of their total areas? Of their volumes ? 859. What formula expresses the total area of a cone of revolution whose altitude is equal to the radius of its base? 860. What formula expresses the volume of the same cone? 861. What is the ratio of the volume of the same cone to the volume of a regular tetrahedron having the same altitude? 862. What should be the altitude of such a cone, that its lateral area may be 100 sq. in. ? 863. What should be the altitude of such a cone, that its volume may be 1000 cu. in. ? 864. What is the lateral area and the total area of a frustum of a cone of revolution whose altitude is 20 in., and the diameters of whose bases are 6 in. and 14 in. respectively? 865. What is the volume of the same frustum ? 866. The diameters of the bases of a frustum of a cone of revolution are 10 in. and 16 in. respectively; its volume is 575 cu. in. What is its altitude? 867. How far from the base must a cone, whose altitude is 16 in., be cut by a plane so that the frustum shall be equivalent to one half the cone ? 868. What is the ratio of the lateral surfaces of a right circular cylinder and a right circular cone of the same base and altitude, if the altitude is three times the radius of the base? 869. The diameter of a right circular cylinder is 10 ft., and its altitude 7 ft. What is the side of an equivalent cube? 870. The altitude of a cone of revolution is 15 in., and the radius of its base 5 in. What should be the diameter of a cylinder of revolution having the same altitude and lateral area? 871. What should be the ratio of the exterior to the interior diameter of a hollow cylinder of revolution, so that it shall contain one half the volume of a solid cylinder of the same dimensions? 872. In order that a cylindrical tank with a depth of 12 ft. may contain 2000 gal., what should be its diameter ? 873. How many cubic inches of iron would be required to make that tank, its walls being one third of an inch thick? SPHERES. PROPOSITION VII. THEOREM. 699. The area of the surface generated by a straight line revolving about an axis in its plane, is measured by the product of the projection of that line upon the axis by the circumference of the circle whose radius is the perpendicular from the axis to the mid point of the line. Given: ab, the projection upon XY of AB revolving about XY, and OP to AB at its mid point, and meeting X Y in 0; To Prove: Area generated by AB is equal to ab × 2 π · Draw PD to XY, and AC to XY. ОР. Since the surface generated by AB is the lateral surface of a frustum of a cone, (576) If AB meets XY, the surface generated is still a conical surface whose area=ab × 2 π · OP, as follows from Prop. III. If AB is parallel to X Y, the surface generated is a cylindrical surface whose area = ab × 2 π · OP, as follows from Prop. I. 700. The area of the surface of a sphere is measured by the product of its diameter by the circumference of a great circle. Given S, the surface of a sphere generated by the revolution of the semicircle ABCDE about the diameter AOE, where O A equals R; To Prove: Area of S is equal to AE × 2π · R. Inscribe in the semicircle a regular semipolygon AB... E, of any number of sides, and draw Bb, Cc, Dd, is to AE. From O draw OP to AB. and is equal to each of the Is chords BC, CD, DE (182). Then OP bisects AB (172), drawn from 0 to the equal Now area AB = Ab × 2 π· OP, area BC= bc × 2 π • OP, (699) = area CD cd × 2 π. OP, etc.; .. if s' denote the surface generated by the semipolygon, OP. S' = (Ab + bc + cd + dE) × 2 π · O P = A E × 2 π · Conceive the number of sides of the semipolygon to be indefinitely increased. Then, as OP has for limit R, the semipolygon for limit the semicircle, and s' for limit s, 701. COR. 1. The surface of a sphere is equivalent to four 702. COR. 2. The areas of the surfaces of two spheres are to each other as the squares of their radii or diameters. 703. DEFINITIONS. A zone is a portion of the surface of a sphere included between two parallel planes. The altitude of the zone is the perpendicular distance between the parallel planes. The bases of the zone are the circumferences of the bounding circles. The zone is called a zone of one base, if one of the parallel planes is tangent to the sphere; that is, a zone of one base is the surface cut off by a plane. 704. COR. 1. The area of a zone is measured by the product of its altitude by the circumference of a great circle. For (see diagram for Prop. VIII.) the area of the zone generated by the revolution of the arc BC= bc × 2 π· R. 705. COR. 2. Zones on the same sphere, or on equal spheres, are to each other as their altitudes. 706. COR. 3. The area of a zone of one base is measured by the area of the circle whose radius is the chord of the generating arc. For the arc AB generates a zone of one base whose area is EXERCISE 874. The diameter of a sphere being 1 ft., what is the area of a great circle of that sphere, and of the sphere itself? 875. If the area of its surface is 400 sq. in., what is the diameter of the sphere? 876. What is the ratio of the surfaces of two spheres whose radii are 10 in. and 12 in. respectively? 877. What is the area of a zone of a sphere 12 in. in diameter, the altitude of the zone being 3 in. ? 878. What fraction of the diameter of a sphere should the altitude of a zone be so as to contain 1th of the surface? n |