104. The angles of a spherical pentagon are 90°, 100°, 120°, 140°, and 150°, and the radius of the sphere is 7 in. Find the area of the pentagon. 105. An equiangular spherical hexagon on a sphere of radius 5 ft. has an area of 9 sq. ft. Find the angles of the hexagon. 106. The volume of a sphere is 80 cu. ft. Find the volume of a triangular spherical pyramid, the angles of whose base are 103°, 112°, and 127°. 107. If the radius of a sphere is 6 in., what is the volume of a triangular spherical pyramid, the angles of whose base are 84°, 93°, and 108° ? 108. Find the volume of a quadrangular spherical pyramid, the angles of whose base are 107°, 118°, 134°, and 146°, the diameter of the sphere being 12 in. 109. The dihedral angles of a trihedral angle are 100°, 65°, and 87°. The trihedral angle is closed by a portion of a sphere whose radius is 4 in. and whose centre is at the vertex of the trihedral angle. Find the area of this portion of the sphere. 110. The perimeter of a certain spherical triangle is equal to one half of a great circle of the sphere on which it lies. What part of the area of the sphere is the area of the polar triangle? What part of the volume of the sphere is cut out by the planes through the centre of the sphere and the sides of the second triangle? 111. The angles of a spherical triangle on a sphere of radius r are 86°, 113°, and 125°. Find the altitude of a zone on this sphere which has the same area as the triangle. 112. On the same sphere are an equilateral spherical triangle, each of whose angles are 93°, and a lune whose angle is 75°. Find the ratio of the areas of these two figures. 113. What part of the volume of a sphere is the volume of a spherical segment of one base whose altitude is one-half the radius of the sphere. 114. A spherical segment of one base is cut from a sphere of radius r, and the radius of the base of the segment is a. Find the volume of the segment. 115. If the radius of a sphere is 12 ft., find the volume of the spherical sector whose base is a zone with an altitude of 2 ft. 116. A sphere floats in water so that three fourths of its surface is submerged. What part of its volume is submerged? 117. How many gallons of water must be poured into a hemispherical bowl of radius 14 in. in order that the water may be 14 in. deep? 118. The distance of a plane from the centre of a sphere is one third the radius of the sphere. Find the ratio of the volumes of the two solids into which the sphere is divided by this plane. 119. What parallel of latitude possesses the property that one fourth of the earth's surface lies to the north of it? What part of the earth's volume is north of the plane of this parallel? 120. The radius of a sphere is 7 ft. The volume of a segment of one base is one half the volume of the spherical sector of which it forms a part. Find its volume. 121. The volume of a segment of one base of a sphere of radius r is equal to the volume of a sphere whose radius is equal to the altitude of the segment. Find the altitude. 122. Find the volume of a spherical segment if the diameter of each base is 8 ft. and the altitude of the segment is 6 ft. 123. A 90° segment of a circle of radius r is revolved about a line through the centre parallel to its chord. Find the volume thus generated. 124. What part of the volume of the earth is the volume of the segment included between the plane of the equator and the plane of the parallel of 30° north latitude? 125. A sphere of radius 13 in. has a cylindrical hole bored through it, the axis of the cylinder passing through the centre of the sphere and the radius of the cylinder being 5 in. Find the total area and the volume of the solid which is left. 126. The surface of a sphere is 31 sq. ft. Find the surface of another sphere having three times the volume of the former. 127. The radius of a sphere is r. Find the radius of a sphere whose area is twice that of the given sphere. 128. The radius of a sphere is r. Find the radius of a sphere whose volume is twice that of the given sphere. 129. Find the ratio of the areas of two mutually equiangular spherical triangles, one on a sphere of radius 1 ft., the other on a sphere of radius 2 ft. |