699. PROPOSITION XXXI. Draw a semicircle. Take any arc, A B, and to the extremities draw radii AC, BC, and draw the Is to the diameter, B E and A D. Let this figure be revolved about the diameter F G. If we add the volume generated by C B E to that generated by A C B, and then deduct the volume generated by A CD, what solid will remain? Can you show how to find the volume of a spherical segment? Sug. 1. Call the radius of the sphere r, radius of upper base of segment, radius of lower baser", altitude of segment D E, h, the volume of the segment v. Sug. 2. Find an expression for the cone generated by CBE. What represents the volume of the sector generated by AC B? Find an expression for the cone generated by A D C. 2 Sug. 3. Find the value of C E in terms of r and r', -2 Find the value of D C in terms of r and ". Can you show that . (1) υ } π [2 r2 (C E — C D) + (›2 — CE) C E — (y2 — CD) CD]? (2) v = 18 π [2 r2 (CE — CD) + r2 (CE -- CD) — (CE® - CD) ]? (3) v — } π h [3 r2 (CE+CE CD+C D')]? = 3 2 2 (4) (C E — CD)2 = CE-2CE CD+C D2 = h2 ? (5) 3 C E + 3 C D = h2 + 2 CE + 2 CF 2 2 CD+2 C D ? h (6) C E2 + C E⋅ CD + C D2 = ‡ ( C E2 + C D′ ) — 1⁄2 ? h9 2 ]. (7) .. from (3) v = § π h [ } (r'2 + r''2) + 12 h2. Review these steps carefully until you thoroughly under stand this proposition. Write Prop. XXXI. 700. Cor. If the segment be of one base, as that gen erated by F B E, show that v = }} = r2 2 h + } π h3). 701. PROPOSITION XXXII. Given: The E cylinder A E B CD with the inscribed sphere H KF G. Call A K or O K, r, the radius of the sphere and the radius of the base of the cylinder. Can you prove (1) that the surface of the cylinder to surface of the sphere :: 3 : 2, and (2) that the volume of the cylinder the volume of the sphere 3 2? Sug. 1. Express the factors in terms of and A D or in terms of and r. Note. The celebrated geometer Archimedes discovered this interesting theorem. Read his biography. 702. Cor. Suppose a cone to have the same base and altitude as the cylinder circumscribing the sphere. Can you show that the cylinder sphere: cone :: 3:2:1? Sug. Express volume of cone in terms of and r. EXERCISES. 538. What is the convex surface of the largest cylinder that can be made from a cube whose edge is 14 feet? 539. A cube of steel weighs 9 pounds. What will be the largest bicycle cone that can be turned from it? 1 cubic inch of steel weighs 4 53+ ounces. 540. If the edge of a regular tetraedron is 4, can you show that the radius of the inscribed and circumscribed spheres equals 1/3 V6 and Ve? Compare the volume of a cube inscribed in a sphere with that circumscribed about the sphere. 541. Let an equilateral triangle revolve about an altitude. Compare the convex surface of the cone generated with the surface of the sphere generated by the inscribed circle. 542. In Ex. 541 compare the volumes generated. 543. Given a cone the radius of whose base equals the radius of a sphere, and whose altitude equals the diameter of the sphere. Can you prove the volume to each other as 1 : 2? 544. On the same sphere, or on equal spheres, zones of equal altitudes are equal in area. 545. How many square feet in a spherical triangle whose angles are 200°, 156°, 95°, the radius of the sphere being 15 inches? 546. How many sheets of tin 20 inches by 28 inches are required to cover a globe 32 inches in diameter. 547. Compare the volume of the moon with that of the earth, assuming the diameter of the moon to be 2,000 miles and that of the earth 8,000 miles. 548. What is the cost of cementing the bottom and curved surface of a cylindrical cistern 10 feet deep and 8 feet in diameter, at 20 cents per square yard? 549. What is the ratio of the surface of a sphere to the entire surface of its hemisphere? 550. From Ex. 547 compare the amount of light reflected to a given point in space equally distant from both the earth and moon. 551. Prove how the areas of two zones on the same or equal spheres are related. 552. Let and " be the radii of two spheres. How are they related? 553. The altitude of two zones on a given sphere are 3 inches and 8 inches. What is the ratio of their surfaces? 554. What is the polar of a trirectangular triangle? 555. A spherical triangle is to the surface of a sphere as the spherical excess is to eight right angles. 556. All triangles on the same or equal spheres having equal angle-sums are equivalent. 557. What is the volume of a spherical segment of one base, whose altitude is 6 cm. and the radius of whose sphere is 20 cm. 558. Compare the surface of a sphere of diameter d with the convex surface of a circumscribed cylinder. 559. A circular sector has its central angle 30° and radius 12 dm. If this sector is revolved about a diameter perpendicular to one of its radii, find the volume generated. |