PROBLEM I. To find the Convex Surface of a Cylinder. ART. 50. Rule. Multiply the circumference of the base by the length of the cylinder, and the product will be the convex surface required: To this add the areas of the two ends when the entire surface is required. The truth of this rule will easily appear by covering a cylinder with a paper, and then spreading it out into a plane. This plane will form a parallelogram, whose length is the same as the length of the cylinder, and whose breadth is the same as the perimeter or circumference of the cylinder. And by Art. 4, the area of the parallelogram thus formed, must be equal to the length multiplied into the breadth. Ex. 1. What is the convex surface of a right cylinder, whose length is 23 feet, and the diameter of its base 3 feet ? OPERATION. First, by Art. 16 find circumf. of base. Then, 3×3.14159-9.42477. 9.42477×23=216.76971=surface. Ex. 2. Required the entire surface of a cylinder whose length is 20 feet, and the diameter of whose base is 2 feet? Ans. 131.95 ft. Ex. 3. What is the entire surface of a cylinder whose length is 82, and circumference of the base 71? PROBLEM II. Ans. 6624.32. To find the Solidity of a Cylinder. ART. 51. Rule.-Multiply the area of the base by the height, and the product will give the solid contents. 1 Ex. 1. What is the solidity of a cylinder, the diameter of whose base is 16 feet, and its height 28 feet? OPERATION. First, find the area of the base by Art. 21. 16-256. Then, 256x.7854-201.0624= area of the base. Then, 201.0624×28=5629.7472=solid contents. Ex. 2. What is the solidity of a cylinder, whose height is 424, and the circumference of its base 213? Ans. 1530837. Ex. 3. What is the solidity of a cylinder whose length is 5 feet, and the diameter of the end 2 feet? Ans. 15.780 solid ft. Ex. 4. Required the solidity of a cylinder the circumference of whose base is 40 feet, and the height 20 feet. Ans. 254.656. solid ft. Ex. 5. The Winchester bushel is a hollow cylinder, 18 inches in diameter and 8 inches deep: what is its capacity? -2 First, the area of the base=18.5×.7854=268.8025. Then, 268.8025×8=2150.42.00=capacity in cubic inches. NOTE. By this rule, every sealer of Weights and Measures may determine the exact capacity of any measure submitted to his inspection. And so any one may test the accuracy of any measure, whether dry or liquid, by reducing its capacity to cubic inches and dividing by the number of cubic inches contained in such measure. The divisor for any measure may be found in the Table of Weights and Measure, page 9. PROBLEM III. To find the Convex Surface of a Cone. ART. 52. Rule. Multiply the perimeter of the base by the slant height, and the product will be the surface, to which add the area of the base when the entire surface is required. Ex. 1. The diameter of the base of a right cone is 3 feet, and the slant height is 15 feet; what is the convex surface? OPERATION. First, 3×3.14159=9.42477=circ. of base. Ex. 2. The diameter of the base of a cone is 4.5 feet, and the slant height 20 feet; what is the entire surface? Ans. 157.276 sq. ft. Ex. 3. If the axis of a cone be 16 and the circumference of the base 75.4, what is the whole surface? First, find the diameter, of which will be the leg of the triangle; then the square root of the sum of the squares of the perpendicular and base will be the slant height? Ans. 1206.4. Ex 4. The circumference of the base of a cone is 10.75, and the slant height 18.25; what is the surface ? PROBLEM IV. Ans. 98.0937. To find the Solidity of a Cone. ART. 53. Rule.- Multiply the area of the base by of the height, and the product will be the solidity. We have already seen that the solidity of the cylinder is equal to the product of the area of the base into the perpendicular height, (Art. 51.) Now, it may be proved by Geometry, that if a cone and a cylinder have the same base and altitude, the cone is of the cylinder. Consequently, the solidity of the cone is equal to the area of the base into of the height. Ex. 1. What is the solidity of a right cone whose height is 10 feet, and the circumference of the base is 9 feet ? We here multiply the area of the base OPERATION. First, 92-81, and 10÷3=3= height. by of the height, Now, 81x.7854-63.6174, area of base. Then, 63.6174×3=222.6609, Ans. and the product is the solidity. Ex. 2. What is the solidity of a cone whose base is 3 feet 6 inches in diameter, and the perpendicular height 9 feet? Ans. 28.86345 cubic ft. Ex. 3. Required the solidity of a cone whose height is 663, and the diameter of whose base is 101 ? Ans. 1770622. Ex. 4. What is the solidity of a cone the circumference of whose base is 40 feet, and the height 50 feet ? Ans. 2122.1333 solid ft. PROBLEM V. To find the Surface of a Frustrum of a Cone. ART. 54. Rule.-Add together the circumferences of the two ends, and multiply the sum by the slant height of the frustrum; the product will be the convex surface: to which add the areas of the two bases when the entire surface is required. This rule is precisely the same as that for a frustrum of a pyramid (Art. 41;) and if a cone be considered as a pyramid of an infinite number of sides, it is equally applicable to the measurement of the frustrum of a cone. Ex. 1. What is the convex surface of the frustrum of a cone, the circumference of the greater base being 30 feet, and of the smaller, 10 feet, the slant height being 20 feet? OPERATION. 10 40=circum. of the two ends. half slant height= 10 400 sq. ft. Ex. 2. Required the convex surface of the frustrum of a cone, the diameter of the greater base being 44, and of the smaller 33, and the slant height 84? Ans. 10159.8. Ex. 3. What is the entire surface of the frustrum of a cone whose slant height is 20 feet, and the diameters of the bases 8 and 4 feet? Ans. 439.824 sq. ft. PROBLEM VI. To find the Solidity of the Frustrum of a Cone. ART. 55. Rule.-I. Add to the areas of the two ends of the frustrum the square root of their product. II. Multiply this sum by of the perpendicular height, and the product will be the solidity. |