From ART. 145 we have the following RULE. To find the convex surface of a cylinder, multiply the circumference of the base by the altitude. WRITTEN EXERCISES. 1. Find the convex surface of a cylinder whose altitude is 10 ft. and the radius of the base 4 ft. 2. Find the convex surface of a cylinder whose altitude is 50 ft. and the diameter of the base 6 ft. 3. Find the entire surface of a cylinder whose altitude is 60 ft. and the diameter of the base 5 ft. 4. Find the entire surface of a cylinder whose altitude is 30 ft. and the radius of the base 7 ft. 257. We have found by ART. 253 that the volume of a prism equals the area of the base multiplied by the altitude. This will be true when the prism has an infinite number of sides and coincides with the cylinder. Hence the RULE. To find the volume of a cylinder, multiply the area of the base by the altitude. WRITTEN EXERCISES. 1. Required the contents of a cylinder 80 ft. long and 4 ft. in diameter. 2. Find the contents of a cylinder whose altitude is 4 rd. and the radius of the base 14 in. 3. Required the contents of a log whose length is 15 ft. and the diameter of the base 8 ft. 258. The Cone. A Cone is a round body whose base is a circle, and whose convex surface tapers uniformly to a point called the vertex. The altitude of a cone is the perpendicular distance from the vertex to the base; as, CO. The slant height is the distance from the vertex to the circumference of the base; as, CA. 259. We have found by ART. 254 that the convex surface of a pyramid equals the perimeter of the base multiplied by one-half the slant height. This will be true when the pyramid has an infinite number of sides and coincides with the cone. Hence the RULE. B To find the convex surface of a cone, multiply the circumference of the base by one-half the slant height. WRITTEN EXERCISES. 1. Find the convex surface of a cone whose slant height is 36 in. and the radius of the base 8 in. 2. Find the convex surface of a cone whose slant height is 60 ft. and the diameter of the base 20 ft. 3. A tent in the form of a cone has a slant height of 18 ft. and a diameter of 30 ft.; how many square yards of muslin are required to make it? 4. Find the entire surface of a cone whose slant height is 66 ft. and the radius of the base 22 ft. 5. Find the entire surface of a cone whose altitude is 15 in. and the radius of the base 8 in. 260. We have found by ART. 255 that the volume of a pyramid equals the area of the base multiplied by one-third of the altitude. This will be true when the pyramid has an infinite number of sides and coincides with the cone. Hence the RULE. To find the volume of a cone, multiply the area of the base by one-third of the altitude. WRITTEN EXERCISES. 1. Find the volume of a cone whose altitude is 12 ft. and the radius of the base 4 ft. 2. Find the volume of a cone whose altitude is 63 ft. and the diameter of the base 20 ft. 3. What is the volume of a cone whose slant height is 17 in. and the diameter of the base 16 in.? 4. Find the volume of a cone whose altitude is 72 ft. and the radius of the base 20 ft. 261. The Frustum of a Pyramid or Cone. The Frustum of a Pyramid is the part of the pyramid which remains after cutting off the top by a plane parallel to the base. 262. The Frustum of a Cone is the part of a cone which remains after cutting off the top by a plane parallel to the base. RULE. To find the convex surface of a frustum, multiply the sum of the perimeters or circumferences of the bases by one-half the slant height. WRITTEN EXERCISES. 1. Find the convex surface of the frustum of a square pyramid whose slant height is 24 ft., and the side of the lower base 10 ft. and upper base 6 ft. 2. Find the convex surface of the frustum of a cone whose slant height is 42 ft., and the radii of the bases 12 and 4 ft., respectively. 3. Find the entire surface of the frustum of a cone whose slant height is 50 ft., and the radii of the bases 10 and 5 ft., respectively. 4. Find the entire surface of the frustum of a triangular pyramid whose slant height is 40 in., and the sides of the upper base 4 in. and of the lower base 10 in. RULE. 263. To find the volume of a frustum, take the sum of the areas of the two bases, to which add the square root of their product, and multiply this sum by one-third of the altitude. WRITTEN EXERCISES. 1. Find the volume of the frustum of a square pyramid the sides of whose bases are 3 ft. and 6 ft. and altitude 12 ft. The volume = (32 + 62 + √⁄32 × 62) 12 = (9 + 36 +18) 4 = 252 cu. ft. 2. Find the contents of the frustum of a cone the radii of whose bases are 3 and 4 ft. and altitude 18 ft. The contents = (π 32 + π 42 + Vπ2 32 × 42) 1o = (9 π + 16 π + 12 π) 6 697.4352 cu. ft. 3. Find the contents of the frustum of a square pyramid the sides of whose bases are 2 and 8 ft. and altitude 15 ft. 4. Find the contents of a log the radii of whose bases are 2 and 3 ft. and length 60 ft. 5. How many cubic feet in the frustum of a cone the radii of whose bases are 12 and 8 ft. and altitude 27 ft.? 6. Find the contents of the frustum of a triangular pyramid the sides of whose bases are 4 and 6 ft. and altitude 21 ft. The surface of a sphere is equal to the square of the diameter, or four times the square of the radius, multiplied by 3.1416. WRITTEN EXERCISES. 1. Find the surface of a sphere whose radius is 5 in. 2. Find the surface of a sphere whose radius is 12 in. 3. Assuming the earth to be a sphere 7960 miles in diameter, how many square miles in its surface? 4. What will it cost to plate with gold a sphere 18 in. in diameter at $40.50 a square foot? 5. What will it cost to tin the hemispherical dome of an observatory 40 ft. in diameter, at 10 cents a square foot? |