64. A circular sector whose central angle is 45° and radius 12 revolves about a diameter perpendicular to one of its bounding radii. Find the volume of the spherical sector generated. 65. Given the area of the surface of a sphere, S, to find its volume. 66. Given the volume of a sphere, V, to find the area of its surface. 67. A right triangle, whose legs are a and b, revolves about its hypotenuse as an axis. Find the area of the entire surface, and the volume, of the solid generated. 68. The parallel sides of a trapezoid are 12 and 26, B respectively, and its non-parallel sides are 13 and 15. Find the volume generated by the revolution of the trapezoid about its longest side as an axis. (Represent BE by x.) 69. An equilateral triangle, whose altitude is h, revolves about one of its altitudes as an axis. Find the area of the surface, and the volume, of the solids generated by the triangle, and by its inscribed circle. (Ex. 21, p. 151.) 70. Find the lateral area and volume of a cylinder of revolution, whose altitude is equal to the diameter of its base, inscribed in a cone of revolution whose altitude is h, and radius of base r. (Represent altitude of cylinder by x.) 71. Find the lateral area and volume of a cylinder of revolution, whose altitude is equal to the diameter of its base, inscribed in a sphere whose radius is r. 72. An equilateral triangle, whose side is a, revolves about a straight line drawn through one of its vertices parallel to the opposite side. Find the area of the entire surface, and the volume, of the solid generated. (The solid generated is the difference of the cylinder generated by BCHG, and the cones generated by ABG c and ACH.) H F 73. The outer diameter of a spherical shell is 9 in., and its thickness is 1 in. What is its weight, if a cubic inch of the metal weighs lb.? (3.1416.) 74. Find the diameter of a sphere in which the area of the surface and the volume are expressed by the same numbers. 75. A regular hexagon, whose side is a, revolves about its longest diagonal as an axis. Find the area of the entire surface, and the volume, of the solid generated. 76. The sides AB and BC of rectangle ABCD are 5 and 8, respectively. Find the volumes generated by the revolution of triangle ACD about sides AB and BC as axes. 77. The sides of a triangle are 17, 25, and 28. Find the volume generated by the revolution of the triangle about its longest side as an axis. (§ 324.) 78. A frustum of a circular cone is equivalent to three cones, whose common altitude is the altitude of the frustum, and whose bases are the lower base, the upper base, and a mean proportional between the bases of the frustum. (§ 660.) 79. The volume of a cone of revolution is equal to the area of its generating triangle, multiplied by the circumference of a circle whose radius is the distance to the axis from the intersection of the medians of the triangle. (§ 140.) 80. If the earth be regarded as a sphere whose radius is R, what is the area of the zone visible from a point whose height above the surface is H? (§ 271, 2.) 81. The sides AB and BC of acute-angled triangle ABC are √241 and 10, respectively. Find the volume of the solid generated by the revolution of the triangle about an axis in its plane, not intersecting its surface, whose distances from A, B, and C are 2, 17, and 11, respectively. B H B R A D F 82. A projectile consists of two hemispheres, connected by a cylinder of revolution. If the altitude and diameter of the base of the cylinder are 8 in. and 7 in., respectively, find the number of cubic inches in the projectile. (π = 3.1416.) A 83. A segment of a circle, whose bounding arc is a | quadrant, and whose radius is r, revolves about a diameter parallel to its bounding chord. Find the area of the entire surface, and the volume, of the solid generated. x 84. If any triangle be revolved about an axis in its plane, not parallel to its base, which passes through its vertex without intersecting its surface, the volume of the solid generated is equal to the area of the surface generated by the base, multiplied by one-third the altitude. (Compare § 666. Case I., Figs. 1 and 2, when a side coincides with the axis; there are two cases according as AD falls on BC, or BC produced. Case II., Fig. 3, when no side coincides with the axis; prove by Case I.) B M E B M E 85. If any triangle be revolved about an axis which passes through its vertex parallel to its base, the volume of the solid generated is equal to the area of the surface generated by the base, multiplied by one-third the altitude. (Compare Ex. 72, p. 383. There are two cases according as AD falls on BC, or BC produced.) LL D F F A N Fig. 1. Fig. 2. D 86. Find the area of the surface of the sphere circumscribing a regular tetraedron, whose edge is 8. (Draw lines DOE and AOF 1 to ABC and BCD, respectively.) B E APPENDIX. PROOF OF STATEMENT MADE IN ELEVENTH LINE, PAGE 201. 683. Theorem. The circumference of a circle is shorter than the perimeter of any circumscribed polygon. Given polygon ABCD circumscribed about a O. F To Prove circumference of O shorter than perimeter ABCD. D E Proof. Of the perimeters of the O and of its circumscribed polygons, there must be one perimeter such that all the others are of equal or greater length. But no circumscribed polygon can have this perimeter. B For, if we suppose polygon ABCD to have this perimeter, and draw a tangent to the O, meeting CD and DA at points E and F, respectively, then since str. line EF is broken line EDF, the perimeter of circumscribed polygon ABCEF is < perimeter ABCD. Hence, the circumference of the O is cumscribed polygon. the perimeter of any cir PROOFS OF THE LIMIT STATEMENTS OF § 640. 684. We assume the following: A portion of a plane is less than any other surface having the same boundaries. 685. Theorem. The total surface of a circular cylinder is less than the total surface of any circumscribed But no circumscribed prism can have this total surface. For suppose prism AC' to have this total surface; and let BCDFE - E' be a circumscribed prism, whose face EF" intersects faces AB' and AD' in lines EE' and FF", respectively. Now, face EF" is < sum of faces AE', AF', AEF, and A'E'F'. (§ 684) Whence, total surface of prism BCDFE – E' is total surface of prism AC'. Then, total surface of cylinder EG is < total surface of any circumscribed prism. PROOFS OF THE LIMIT STATEMENTS OF § 640. 686. Let L denote the lateral edge, H the altitude, S and s the the lateral areas, V and v the volumes, E and e the perimeters of rt. sections, and B and b the areas of the bases of the circumscribed and inscribed prisms, respectively; also, S' the lateral area of the cylinder, V' its volume, E' the perimeter of a rt. section, and B' the area of the base. 1. We have, S+2 B>S' + 2 B'. .. S+2(B - B') > S'. ($ 685) Again, the total surface of the inscribed prism is the total surface of the cylinder. :. S' + 2 B' > s +2b, or S'> s + 2 (b − B'). S + 2 (B − B') > S' > s + 2 (b B'). ($ 684) Then, Now if the number of faces of the prisms be indefinitely increased, B- B' and b - B' approach the limit 0. (§ 363, II) Again, the difference between the perimeters of the bases of the prisms approaches the limit 0. (§ 363, I) Then, the total surface of the circumscribed prism continually decreases, but never reaches the total surface of the inscribed prism; and the total surface of the inscribed prism continually increases, but never reaches the total surface of the circumscribed prism. (§ 684) Then, the difference between S+ 2 B and s + 2 b can be made less than any assigned value, however small. Whence, S + 2 B − (s + 2 b), or S − s + 2 (B — b), approaches the limit 0. But Bb approaches the limit 0. (§ 363, II) Then, S' is intermediate in value between two variables, the difference between which approaches the limit 0. |