Multiply the area of its base by one-third of its perpendicular altitude. The convex surface of a cone is the sector of a circle, whose arc is the circumference of the base, and whose radii are equal to the slant, or slanting height of the cone. Therefore, To find the surface of a cone : Multiply the circumference of the base by one-half the slant height, and to the product add the area of the base. In any cone or pyramid, a section or plane parallel to the base is similar to the base, and the areas of the two planes are to each other as the squares of their respective distances from the vertex. To find the solidity of a cone by the sliding rule:-Place one-third of its altitude over the gauge points for the cylinder, and over the diameter of the base found on D, will be found its solidity on C. See 36, page 119. EXAMPLES. 1. The altitude of a right cone is 30 feet, and the diameter of its base 6 feet; what is its solidity? Ans. 282.74 feet. 2. The altitude of a right cone is 12, and the diameter of its base 10; what is its solidity? Ans. 314.16. 3. Find the solidity of a cone, whose altitude is 20 feet, and the diameter of its base 30 inches. Ans. 32.76. 4. The altitude of a cone is 27 feet, and the diameter of its base 7 feet; find its contents in feet and bushels. Answers, 346 feet, and 278 bushels, nearly. 5. The altitude of a cone is 9 feet, and the diameter of its base 15 inches; find its contents in ale and wine gallons. ale, and 27 wine gallons. Answers, 22 6. The altitude of a cone is 25 feet, and the circumference of its base 20 feet; what is its solidity? Ans. 265.25 feet. 7. What is the surface of a cone, whose altitude is 18 feet, and the diameter of its base 5 feet? Ans. 162.364 feet. 8. The slant height of a cone is 40 feet, and the diameter of its base 9 feet; what is its surface? Ans. 629.11 feet. 9. The altitude of a cone is 120 feet, and the circumference of its base 30 inches; what is the length of an inch ribbon, which will wind round the cone from its base to its vertex, and leave a space of 5 inches between the several flexures? Ans. 100 yards, nearly. T42. FRUSTUM OF A CONE. To find the solidity of a frustum of a cone : 1. Complete the solid, and proceed as directed in T 40. 2. Or, Multiply the diameter of the greater base by the diam eter of the less, and to the product add one-third of the square of the difference of the two diameters, and multiply the sum by 0.7854, and the product will be the mean area, which, multiplied by the altitude of the frustum, will give the solidity. 3. Or, Multiply the sum of the squares and the product of the two diameters by .2618, and the product will be the mean area. 4. Or, To the squares of the circumferences of the two ends add the product of the circumferences, and multiply the sum by the height of the frustum, and this product by .02652, and the result will be the solidity. To find the area of the convex surface of a conic frustum : Multiply half the sum of the circumferences of the two ends by the slant height. The diameters of the two ends of a conic frustum and its solidity being given, to find its altitude: Divide the solidity by its mean area, and the quotient will be the altitude. EXAMPLES. 1. What is the solidity of a frustum of a cone, whose altitude is 10 feet, and the diameters of its ends 2 and 4 feet? Ans. 73.3 feet. 2. How many cubic feet are contained in a ship's mast, whose length is 72 feet, and the diameters of its ends 1 foot and 14? Ans. 89.5356. 3. How many cubic feet are contained in a cask, which is composed of two equal and similar conic frustums, united at their greater ends, its bung diameter being 14 inches, its head diameter 10 inches, and its length 20 inches? Ans. 1.34838. 4. A ship's mast is 60 feet long, and the diameters of its ends 12 and 30 inches; find its superficial and solid contents. Answers, 335.562 square, and 153.15 solid feet, 5. The bung diameter of a cask composed of two conic frustums, is 26 inches, the head diameters 20 inches each, and its length 30 inches; required its contents in ale and wine gallons. Answers, 44.33, and 54.33. 6. How many wine gallons will a tub hold, whose diameters are 30 and 40, and its depth 60 inches? Ans. 251.6 gallons. Find the mean diameter, and then place the length of the frustum over the gauge points for the cylinder, and over the mean diameter, found on D, will be found its contents on C. The mean diameter may be found sufficiently near for most practical purposes, by adding .52 of the difference of the two diameters to the less diameter. 7. What is the convex surface of a conic frustum whose slant height is 10 feet, and the circumferences of its two ends, 5 and 15 feet? Ans. 100 feet. 43. THE WEDGE. A wedge is a solid, having a rectangular base, and two opposite sides terminating in an edge. When the edge and base are of equal length, it is called a direct wedge; and when the base is longer than the edge, it is called an indirect wedge. |