Mensuration of the Round Bodies. PROBLEM V. To find the surface of the frustum of a cone RULE. Add together the circumferences of the two bases; and multiply the sum by half the slant height of the frustum; the produc will be the convex surface, to which add the areas of the bases when the entire surface is required (Bk. VI. Th. ix). EXAMPLES. 1. What is the convex surface of the frustum of a cone, of which the slant height is 12 feet, and the circumferences of the bases 8,4 and 6 feet. 2. What is the entire surface of the frustum of a cone, the side being 16 feet, and the radii of the bases 2 and 3 feet? Ans. 292.1688 sq. ft. 3. What is the convex surface of the frustum of a cone, the circumference of the greater base being 30 feet, and of the less 10 feet; the slant height being 20 feet? Ans. 400 sq. ft. 4. Required the entire surface of the frustum of a cone whose slant height is 20 feet, and the diameters of the bases 8 and 4 feet Ans. 439.824 sq. ft. Mensuration of the Round Bodies. PROBLEM VI. To find the solidity of the frustum of a cone RULE. I. Add together the arcas of the two ends and a geometrical mean between them. II. Multiply this sum by one-third of the altitude and the product will be the solidity. EXAMPLES. 1 How many cubic feet in the frustum of a cone whose altitude is 26 feet, and the diameters of the bases 22 and 18 feet? First, 22x.7854-380.134-area of ower base: and 182x.7854=254.47= area of upper base. Then, 380.134 × 254.47=311.018=mean. 26 Then, (380.134+254,47+311.018) x 8195.39 which is the solidity. 3 2. How many cubic feet in a piece of round timber the diameter of the greater end being 18 inches, and that of the less 9 inches, and the length 14.25 feet? Ans. 14.68943. 3. What is the solidity of a frustum, the altitude being 18. the diameter of the lower base 8, and of the upper 4? Ans. 527.7888. 4. If a cask, which is composed of two equal conic frustums joined together at their larger bases, have its bung diameter 28 inches, the head diameter 20 inches, and the length Mensuration of the Round Bodies. 40 inches, how many gallons of wine will it contain, there being 231 cubic inches in a gallon? PROBLEM VII. Ans. 79.0613. To find the surface of a sphere. RULE. Multiply the circumference of a great circle by the diameter, and the product will be the surface (Bk. VI. Th. xxiii). 2. What is the surface of a sphere whose diameter is 24? Ans. 1809.5616. 3. Required the surface of a sphere whose diameter is 7921 miles. Ans. 197111024 sq. miles. What is the surface of a sphere the circumference of whose great circle is 78.54? Ans. 1963.5. 5. What is the surface of a sphere whose diameter s Ans. 5.58506 sq. ft. feet? To find the convex surface of a spherical zone. RULE. 4 Multiply the height of the zone by the circumference of a great circle of the sphere, and the product will be the convex surface (Bk. VI. Th. xxiv) 2. The diameter of a sphere is 12 feet: what will be the surface of a zone whose altitude is 2 feet? Ans. 78.54 sq. ft. 3. The diameter of a sphere is 21 inches: what is the surface of a zone whose height is 41⁄2 inches? Ans. 296.8812 sq. in. 4. The diameter of a sphere is 25 feet and the height of the zone 4 feet: what is the surface of the zone? Ans. 314.16 sq. ft. 5. The diameter of a sphere is 9, and the height of a zone 3 feet: what is the surface of the zone? Ans. 84.8232. PROBLEM IX. To find the solidity of a sphere. RULE I. Multiply the surface by one-third of the radius and the product will be the solidity (Bk. VI. Th. xxv). 2. The diameter of a sphere is 7957.8: what is its solidity? Ans. 263863122758.4778. 3. The diameter of a sphere is 24 yards: what is its solid contents? Ans. 7238.2464 cubic yds. 4. The diameter of a sphere is 8: what is its solidity? Ans. 268.0832. The diameter of a sphere is 16: what is its solidity? Ans. 2144.6656 RULE II. Cube the diameter and multiply the number thus found, by the decimal .5236, and the product will be the solidity. EXAMPLES. 1 What is the solidity of a sphere whose diameter is 20? Ans. 4188.8. 2. What is the solidity of a sphere whose diameter is 6? Ans. 113.09'76. 3. What is the solidity of a sphere whose diameter is 10? Ans 523.6 |
