68. Find the diameter of a sphere in which the area of the surface and the volume are expressed by the same numbers. 69. An equilateral triangle, whose side is a, revolves about one of its sides as an axis. Find the area of the entire surface, and the volume, of the solid generated. 70. 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. 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 cone of revolution whose altitude is h, and radius of base r. 72. 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. 73. 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. 74. If the radius of a sphere is R, find the circumference and area of a small circle, whose distance from the centre is h. 75. 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. ? 76. 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. 77. The sides AB and BC of a rectangle ABCD are 5 and 8, respectively. Find the volumes generated by the revolution of the triangle ACD about the sides AB and BC as axes. 78. 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. 79. The cross-section of a tunnel 24 miles in length is in the form of a rectangle 6 yd. wide and 4 yd. high, surmounted by a semicircle whose diameter is equal to the width of the rectangle. How many cubic yards of material were taken out in its construction? 80. 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. (§ 677.) 81. 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. 82. If the earth be regarded as a sphere whose radius is R, what 270 is the area of the zone visible from a point whose height above the surface is I? 83. The sides AB and BC of an acute-angled triangle ABC, are √241 and 10, respectively. Find the volume 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. 84. 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. 85. A tapering hollow iron column, 1 in. thick, is 24 ft. long, 10 in. in outside diameter at one end, and 8 in. in diameter at the other. How many cubic inches of metal were used in its construction ? 86. 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 generated is equal to the area generated by the base, multiplied by one-third the altitude. 87. If any triangle be revolved about an axis which passes through its vertex parallel to its base, the volume generated is equal to the area generated by the base, multiplied by one-third the altitude. 88. A segment of a circle whose bounding are 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. 89. Find the area of the surface of the sphere circumscribing a regular tetraedron whose edge is 8. |