To find the curve surface of the frustum of a cone. Multiply the sum of the diameters of the two ends by the slant side; then multiply by 11 and divide by 7. Ex. Find the curve surface of the frustum of a cone whose top and bottom diameters are 12 ft. 2 in., and 15 ft. 4 in., and the slant side 17 ft. 6 in. EXAMPLES. I. Find the curve surface of a cone whose diameter is 13 ft. 4 in., and slant side 18 ft. 8 in. Ans. 3914 feet. 2. Find the curve surface of a cone whose diameter is 19 ft. 2 in., and altitude 21 feet. Ans. 695 feet, nearly. 3. Find the curve surface of a cone whose altitude is 1 yards, and diameter yard; and the cost of gilding the same at d. per square inch, including the base. Ans. Surface 163 feet, cost, £3. Is. 1d. 4. The sides of a right-angled triangle are 3, 4, 5; if it revolve about the hypothenuse, find the whole surface generated by the other two sides. Ans. 524 inches. 5. Find the curve surface of the frustum of a cone whose top and bottom diameters are 5 ft. 4 in. and 8 ft. 3 in., and the slant side 10 ft. 6 in. Ans. 224 feet. 6. Find the curve surface of the frustum of a cone whose top and bottom diameters are 15 ft. 4 in., and 26 ft. 8 in., and the depth 23 ft. 9 in. Ans. 1611 feet. XXXII. To find the solidity of a cone. Multiply the square of the diameter by the height; then multiply by 11 and divide by 42. Ex. Find the solidity of a cone whose diameter is 7 ft. 6 in., and height 14 ft. 7 in. To find the solidity of the frustum of a cone. Add the squares of the diameters of the two ends and their product together; multiply the sum by the height of the frustum; then multiply by II and divide by 42. Ex. Find the solidity of the frustum of a cone whose top and bottom diameters are 9 and 16 inches, and the depth 14 inches. 92 = 81 162 = 256 9 × 16 = 144 481 14 6734 6734 6 74074 7 123453 1763 Ans. EXAMPLES. I. Find the solidity of a cone whose diameter is 10 yards, and altitude 11 yards. Ans. 3243 yards. 2. Find the solidity of a cone whose diameter is 7 ft. II in., and slant side 16 ft. 1 in. I Ans. 255.879 feet. 3. Find the solidity of a cone whose diameter is 13 ft. 6 in., and slant side 18 ft. 4 in. Ans. 813.7.2 feet. 4. Find the solidity of the frustum of a cone whose top and bottom diameters are 4 and 7 feet, and the depth 8 feet. Ans. 220. II feet. 5. Find the solidity of the frustum of a cone whose top and bottom diameters are 18 ft. 5 in., and 13 ft. 1 in., and slant side 21 ft. 5 in. Ans. 4181.4 feet. 6. The top and bottom diameters of a tub are 4 ft. 3 in. and 2 ft. 9 in., and the depth 4 ft. 10 in.; how many gallons will it hold? Ans. 294. 7. Find the weight of a conical plug of cast iron whose diameter is 9 inches, and height 15 inches. Ans. 79 lbs. 8. The sides of a right-angled triangle are 1 ft., 2 ft. II in., 3 ft. 1 in.; if it revolve about the hypothenuse, find the solidity and surface of the solid. Ans. 283 feet, 11143 feet. THE SPHERE. O This solid is formed by the revolution of a circle about a diameter. XXXIV. To find the surface of a sphere. Multiply the square of the diameter by 34. Ex. Find the surface of a sphere whose diameter is To find the curve surface of the segment of a sphere. (1) When the diameter of the sphere and the height of the segment are given: Multiply the diameter of the sphere by the height of the segment; then multiply by 34. |