PROP. XXXIII. PROBLEM 601. Given the radii of the bases, and the altitude, of a spherical segment, to find its volume. Given ОМ, АА!' the centre of arc ADB, lines AA', BB' L to diameter Required to express volume of spherical segment generated by the revolution of ADBB'A' about OM as an axis in terms of r, r', h. Solution. 1. Draw lines OA, OB, AB; also, line OCL AB, and line AEL BB'; denote radius OA by R. 2. 3. vol. ADBB'A' : = vol. ACBD + vol. ABB'A'. (1) vol. ACDB = 4. By § 596, vol. OADB = vol. OADB –– vol. OAB. πR2h. 5. By § 594, vol. OAB = area AB × 1 OC = h × 2 πOC × } OC (§ 586) = } πOC2h. 6. Then, vol. ACDB = }} πR2h — } πOC2h = } π(R2 — OC3)h. - 7. By § 253, R2 — OC2 = AC2 = († AB)2 (?) = † AB2. 8. Then, vol. ACDB = { π × ‡ AB2 × h = { πABh = } π(BE2 + AE2)h = } #[(r — r')2 + h2]h. 9. Substitute in (1), by § 511, vol. ADBB'A' = {}; #[(r− r')2 + h2]h + } π(2 r2 + 2 p12 + 2 rr1)h = } π(r2 − 2 rr' + pl2 + h2 + 2 po2 + 2 p12 + 2 rr')h 602. If denotes radius of base, h the altitude, of a spherical segment of one base, its volume is Ex. 43. Find the volume of a cone of revolution inscribed in a sphere whose volume is 4500 π. The axis of the cone is 15 and coincides with a fixed diameter. Can there be more than one such cone? Ex. 44. A plane is drawn through the axis of a cone of revolution. The intersection of the plane and the cone is an equilateral triangle whose side is 12. Find the area of the surface of a sphere inscribed in the cone. Ex. 45. A plane is passed through the axis of a cone of revolution. The intersection of the plane and the cone is an equilateral triangle. Find the ratio of the volume of the sphere inscribed in the cone to the volume of the cone. Ex. 46. The legs of a right triangle are 18 and 12, respectively. The triangle revolves about the hypotenuse as an axis. Find the area of the surface generated. Ex. 47. A hollow iron column in the shape of a frustum of a cone of revolution is 10 feet long; one base is 10 inches in outer diameter, and 8 inches in inner, and the other base is 6 inches in outer diameter, and 4 inches in inner. The column is melted and formed into a bar in the shape of a rectangular parallelopiped, 4 inches wide and 2 inches thick. Find length of bar. Ex. 48. A 10-pound spherical iron cannon ball is 3 inches in diameter. Find the diameter of a 20-pounder. Ex. 49. Measure the circumference of a great circle of any sphere of known diameter. Using the measurement as given, compute the volume of the sphere. Ex. 50. Given a semicircle of radius r, and a chord parallel to the diameter of the semicircle. The radii drawn from the middle point of the diameter to the extremities of the chord form an angle of 60°. The semicircle is revolved about its diameter as an axis. Find the volume of the solid generated by the segment between the chord and the arc. Ex. 51. Given an equilateral triangle ABD, and a light placed at point C, equidistant from A, B and D, and at a distance 12 from plane ABD. A side of ABD is 8 and plane ABD is parallel to plane QR. A'B'D' is the shadow of ABD upon QR, and CO' is perpendicular to A'B'D', meeting ABD at O. The distance from C to QR is 64. Find area A'B'D'. R B INDEX TO DEFINITIONS. Acute angle, § 37. diedral angles, § 378. of a cylinder, § 480. of a frustum of a cone, of a frustum of a pyramid, of a parallelogram, § 103. of a pyramid, § 445. of a trapezoid, § 103. Angle, § 10. at the centre of a regular inscribed in a segment, of a lune, § 566. Angles of a polygon, § 115. of a quadrilateral, § 102. of a triangle, § 21. Angular degree, § 39. Antecedents of a proportion, § 213.! Apothem of a regular polygon, Arc of a circle, § 23. Axis of a circle of a sphere, § 518. Base of a cone, § 495. of a polyedral angle, § 401. of a parallelogram, § 103. of a spherical segment, § 583. of a zone, § 583. Central angle, § 149. circumscribed about a poly- gon, § 152. Circle inscribed in a polygon, | Dimensions of a rectangle, § 279. Commensurable magnitudes, § 181. Distance between two points on Common measure, § 181. tangent, § 151. Complement of an angle, § 40. Complementary angles, § 40. Concave polygon, § 119. Concentric circles, § 146. Cone, § 495. of revolution, § 495. Conical surface, § 495. Constant, § 184. the surface of a sphere, § 524. of a point from a line, of a point from a plane, Dodecaedron, § 410. Edge of a diedral angle, § 378. Consequents of a proportion, § 213. Edges of a polyedral angle, § 401. |