776. COR. In a spherical segment of one base, r1=0. hπι2 ? πh3 + EXERCISES Ex. 1234. Find the volume of a spherical segment, the radii of whose bases are 4 and 5 and whose altitude is 1. Ex. 1235. The volumes of two spheres are to each other as 8 to 125. Find the ratio of their radii. Ex. 1236. The volumes of two spheres are to each other as 125 to 216. Find the ratio of their surfaces. Ex. 1237. Find the radius of a sphere whose surface is equivalent to the sum of the surfaces of two spheres whose radii are 3 and 4 respectively. Ex. 1238. Find the volume of a spherical shell whose exterior radius is 13 and whose thickness is 8. Ex. 1239. Find the radius of a sphere equivalent to the spherical shell in the preceding exercise. Ex. 1240. Find the radius of a sphere equivalent to a cube whose edge is equal to a. Ex. 1241. A cylindrical vessel, 4 in. in diameter, is partly filled with water. Upon immersing a ball the surface of the water rises 1 in. Find the diameter of the ball. Ex. 1242. A sphere whose radius is 2 in. weighs 32 oz. Find the weight of a sphere of the same material whose radius is 3 in. Ex. 1243. Find the volume of a spherical pyramid whose base is an equilateral triangle with its angles equal to 80°, if the radius of the sphere is equal to 10. Ex. 1244. A square whose side is 4 revolves about one of its diagonals. Find the surface and the volume of the generated solid. Ex. 1245. Find the volume of a spherical segment of one base, if its curved surface is 20 and its altitude is 2. Ex. 1246. Find the radius of a sphere whose surface is equivalent to the entire surface of a cube whose edge is equal to 4. Ex. 1247. The edge of a cube is 10 in. Find the diameter of the circumscribed sphere. Ex. 1248. A lune whose angle is equal to 40° is equivalent to a zone on the same sphere. Find the ratio of the altitude of the zone to the radius of the sphere. Ex. 1249. The diedral angles of a spherical pyramid of six sides are 140°. Find the volume of the pyramid if the radius is equal to 10. Ex. 1250. Through a sphere whose diameter is 10 m. a cylindrical hole of 5 m. diameter is bored. Find the volume of the solid if the axis of the cylinder passes through the center of the sphere. Ex. 1251. The surface of a sphere is equivalent to the lateral surface of the circumscribed cylinder. Ex. 1252. Two bi-rectangular spherical triangles are equal if the oblique angles are equal. Ex. 1253. Find the ratio of a sphere to its circumscribed cube. Ex. 1254. The area of a zone on a sphere is 20, its altitude 4. Find the radius of the sphere. Ex. 1255. If the diagonals of a spherical quadrilateral bisect each other, the opposite sides are equal. Ex. 1256. The radius of a sphere is 9 in. Find the volume of a spherical wedge whose angle is equal to 60°. Ex. 1257. Find the radius of a sphere equivalent to a cone of revolution, the radius of whose base is equal to r and whose altitude is equal to h. Ex. 1258. The area of a zone is equal to A, its altitude is equal to h. Find the radius of the sphere. Ex. 1259. The volume of a sphere is numerically equal to one-half its surface. Find the radius. Ex. 1260. The volume of a cylinder of revolution is equal to one-half the product of its lateral surface by the radius of its base. Ex. 1261. How many square miles of the surface of the earth can be seen from a point 1000 miles above the surface, if the earth is supposed to be a perfect sphere whose radius is equal to 4000 miles ? Ex. 1262. If from a point without a sphere a tangent and a secant be drawn, the tangent is the mean proportional between the secant and its external segment. Ex. 1263. If through the line of intersection of two spheres a plane be passed, tangents from a point of the plane to the spheres are equal. Ex. 1264. The radius of a sphere is r, the area of a small circle a. Find its distance from the center. Ex. 1265. The volume of a sphere is V. Find the surface of an equilateral spherical triangle whose angle is equal to 100°. INDEX OF DEFINITIONS PAGE . . 271 . . PAGE 3 5 5 5 73 7 Alternation. . 120 Angle, tetraedral . 314 triedral 306 vertex of vertical. alternate exterior 287 alternate interior complementary 80 corresponding 12 exterior. 358 interior. 61 of polygon. 107 supplementary 219 vertical . 3 Antecedents 4 | Apothem. 74 Arc 251 Area 95 of right triangles 95 Axis of circular cone. 5 of circular cylinder 4 of gular id 350 335 Base of isosceles triangle 262 of pyramid 4 of spherical pyramid . of spherical sector. 335 of triangle 4 of cylinder . . 66 66 . 11 11 PAGE . . 66 66 66 66 . PAGE Cube . 272 305 50 305 271 306 of revolution 306 50 305 358 307 305 directrix of . 305 element of . 305 73 generatrix of 305 327 57 74 5 33, 73 234 74 11 106 73 33, 73 327 90 251 5 edge of . 251 121 faces of 251 plane angle of 252 8 11 313 Distance, from point to line 41 314 from point to plane . 245 313 on surface of sphere 329 314 121 314 125 314 126 313 220 313 57 251 of polyedron. 270 118 Element of conical surface. . 314 91 of cylindrical surface 305 2 164 solids 276 66 . 66 66 nappes of . ::13 . . PAGE 73 12 66 . . . 313 90 270 66 PAGE 118 Maxinum . . Mean proportional 251 Means 262 Median of triangle . 270 Minimum . 2 Minor arc of circle. 2 9 Nappes of cone 4 | Numerical measure 118 315 Octaedron. Octagon Parallel lines. 305 Parallelogram 2 Parallelopiped 2 right 2 rectangular Parallel planes 129 Pentagon 57 Perimeter 57 270 Perpendicular Plane 13 Point 12 Polar distance of circle 7 triangle 270 Pole. 259 | Polygon 90 angles of 121 circumscribed 225 convex diagonal of 91 equiangular 2 equilateral 2 inscribed 2 regular . spherical 111 Polyedral angle 350 Polyedron. 350 convex 17 4 1 . . 76 194 270 |