Ex. 1737. To find the volume of a spherical segment generated by ABCD, if AB = 6 in., BC = 2 in., DC = 4 in., and ▲ B = ≤ C = 90°. Draw OA, and OD. Denote OA by x and OB by y. LB = 90°, x2 = y2 + 36. SOLUTION. Since Since Ex. 1738. Find the volume of a spherical segment, the radii of whose bases are 2 and 5, and whose altitude is 1. Ex. 1739. Find the volume of a spherical segment, the radii of whose bases are 6 and 8, and whose altitude is 2. Ex. 1740. Find the volume of a spherical segment of one base the radius of whose base is 3, and whose altitude is 2. Ex. 1741. Find the volume generated by the rotation of triangle ABC about AB as an axis if AB 14, BC= 15, and CA = 13. Ex. 1742. Find the volume generated by the rotation of trapezoid ABCD about its base AB, if AB 10, BC AD = 5, and BD = 6. = = Ex. 1743. A line XY and a triangle ABC lie in one plane, A lies in XY, and from C, CC' is drawn perpendicular to XY, and similarly BB' 1 XY. Find the volume generated by the rotation of triangle ABC about XY, if AC' = 5, C'B' = 7, AB' = 12, CC 12, and BB' = 5. = Ex. 1744. In the annexed figure ABD is a straight line and B is the center of arc CD. Find the volume generated by the rotation of the entire figure about AD as an axis, if AC = 15, CB 13, and AB = 4. = Ex. 1745. In the same diagram find the volume generated by the rotation of triangle ABC about AB as axis, if AB = 5, BC = 8, and ABC = 120°. B Ex. 1746. The volumes of two spheres are to each other as 8 to 125. Find the ratio of their radii. Ex. 1747. The volumes of two spheres are to each other as 125 to 216. Find the ratio of their surfaces. Ex. 1748. Find the radius of a sphere whose surface is equivalent to the sum of the surfaces of two spheres whose radii are 3 and 4 respectively. Ex. 1749. Find the volume of a spherical shell whose exterior radius is 13 and whose thickness is 8. Ex. 1750. Find the radius of a sphere equivalent to the spherical shell whose exterior radius is 3 and whose thickness is 1. Ex. 1751. If the diameter of the moon is 2160 mi., find its surface and its volume. Ex. 1752. The area of a zone on a sphere is 80, its altitude 4. Find the radius of the sphere. Ex. 1753. Find the radius of a sphere equivalent to a cube whose edge is equal to a. Ex. 1754. A cylindrical vessel, 4 in. in diameter, is partly filled with water. Upon immersing a ball in the water, the surface rises 1 in. Find the diameter of the ball. Ex. 1755. A sphere whose radius is 2 in. weighs 32 oz. Find the weight of a sphere of the same material whose radius is 3 in. Ex. 1756. Find the volume of a spherical pyramid whose base is an equilateral triangle with its angles equal to 80°, if the radius of the sphere is equal to 10. Ex. 1757. A square whose side is 4 revolves about one of its diagonals. Find the surface and the volume of the generated solid. Ex. 1758. Find the volume of a spherical segment of one base, if its curved surface is 20 π and its altitude is 2. Ex. 1759. Find the radius of a sphere whose surface is equivalent to the entire surface of a cube whose edge is equal to 4. Ex. 1760. The edge of a cube is 10 in. Find the diameter of the circumscribed sphere. Ex. 1761. Find the radius of a sphere inscribed in a regular tetrahedron whose edge equals 4 in. Ex. 1762. A lune whose angle is equal to 40° is equivalent to a zone on the same sphere. Find the altitude of the zone if the diameter of the sphere is 18 in. Ex. 1763. The dihedral angles of a spherical pyramid of six sides are 140°. Find the volume of a pyramid if the radius is equal to 10. Ex. 1764. The surface of a sphere is equivalent to the lateral surface of the circumscribed cylinder. Ex. 1765. Find the ratio of a sphere to its circumscribed cube. Ex. 1766. If the diagonals of a spherical quadrilateral bisect each other, the opposite sides are equal. Ex. 1767. The radius of a sphere is 9 in. Find the volume of a spherical wedge whose angle is equal to 60°. Ex. 1768. Find the radius of a sphere equivalent to a cone of revolution, the radius of whose base is equal to r and whose altitude is equal to h. Ex. 1769. The area of a zone is equal to A; its altitude is equal to ǹ. Find the radius of the sphere. Ex. 1770. The volume of a sphere is numerically equal to one half its surface. Find the radius. Ex. 1771. The volume of a cylinder of revolution is equal to one half the product of its lateral surface by the radius of its base. Ex. 1772. What fraction of the surface of a sphere is visible to an observer whose distance from the center equals three times the radius ? Ex. 1773. How many square miles of the surface of the earth can be seen from a point 1000 mi. above the surface, if the earth is supposed to be a perfect sphere whose radius is equal to 4000 mi. ? Ex. 1774. How far from the center of a sphere whose radius is 6 ft. is an observer to whom of the entire surface is visible? 12 Ex. 1775. If from a point without a sphere a tangent and a secant be drawn, the tangent is the mean proportional between the secant and its external segment. Ex. 1776. Through a sphere whose diameter is 10 in. a cylindrical hole of 6 in. diameter is bored. Find the volume of the solid if the axis of the cylinder passes through the center of the sphere. Ex. 1777. The radius of a sphere is r, the area of a small circle a. Find its distance from the center. Ex. 1778. The volume of a sphere is V. Find the surface of an equilateral spherical triangle whose angle is equal to 100°. Ex. 1779. Find the area of the torrid zone if its altitude is of the radius of the earth. Ex. 1780. How many miles from the surface of the earth can of the surface be seen? Ex. 1781. The average pressure of the atmosphere upon each square inch of the surface is 15 lb. Find the total weight of the atmosphere. Ex. 1782. How many bullets of an inch in diameter can be made from a cubic foot of lead? Ex. 1783. A coffee pot is 8 in. high, 4 in. in diameter at the top, and 5 in. in diameter at the bottom. How many cups of coffee, each having a capacity of 10 cu. in., could the pot hold ? Ex. 1784. The places, A, B, and C, on the surface of the earth determine a spherical triangle ABC, whose angles are: A = 50°, B = 61°, C = 71°. Find the area of triangle ABC. APPENDIX TO SOLID GEOMETRY PROPOSITION I. THEOREM 798. A truncated triangular prism is equivalent to the sum of three pyramids whose common base is the base of the prism and whose vertices are the three vertices of the inclined section. Given ABC-HKG, a truncated triangular prism of which ABC is the base. To prove ABC-HGK = H-ABC + K-ABC +G-ABC. Proof. Pass planes through H, B, C, and through H, K, C, forming the pyramids H-ABC, H-BCK, and H-CKG. H-ABC is evidently one of the required pyramids. H-BCK may be read C-HBK. C-HBK = C-AKB may be read K-ABC. (628) .. H-BCK is equivalent to the second of the required pyramids. |