4. The circumference of the base of a cone is 40 feet, and the height 50 feet; required the solidity. Thus, 402x07958 1600 x07958 127.328 = area of base. 50 Then 127-328 x 3 6366.4 =2122-1333 feet. Ans. 5. The circumference of the base of a cone is 10 feet, and the perpendicular altitude 12 feet; what is the solidity? Ans. 31.829 cubic feet. PROBLEM 11. To find the surface of the frustum of a right cone. RULE.-Add together the circumferences of the two bases, and multiply the sum by half the slant height of the frustum, and the product will be the convex surface, to which add the areas of the bases, and the entire surface is acquired. 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? Thus, 84+614-4; half side, 6·20 x 14.4 90 square feet. Ans. Cone. 2. 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 square feet. 3. 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 square feet. PROBLEM 12. To find the solidity of the frustum of a cone. RULE. Add together the areas of the two ends and geometrical mean between them. Multiply this sum by one-third of the altitude, and the product will be the solidity. 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? Thus, 222 x 7854 380.134, area of lower base. = 182 x 7854 = 259.47 area of upper base. 2. What is the solidity of the frustum of a cone, the altitude being 18, the diameter of the lower base 8, and the upper base 4? Ans. 527-7888. 3. 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 feet. 4. What is the solidity of the frustum of a cone, the diameter of the greater end being 4 feet, that of the less end 2, and the altitude 9 feet? Ans. 65.9736 feet. 5. What is the solidity of the frustum of a cone, the circumference of the greater end being 20 feet, and that of the less end 10 feet, and the height 21 feet? Ans. 389-942 feet. PROBLEM 13. The solidity and altitude of a cone being given, to find the diameter. RULE.-Divide the solidity by the product of 7854 and onethird of the altitude, and the square root of the quotient will be the diameter. 1. The solidity of a cone is 16 feet, and the altitude 9 feet; what is the diameter? Thus, 7854 x 3 2.6057 feet. Ans. = 2.3562; 16 ÷ 2·3562 = 2/6.7906 = 2. The solidity of a cone is 18 feet, and the altitude 8 feet; required the diameter. Ans. 2.9315 feet. PROBLEM 14. The solidity and diameter of a cone being given, to find the altitude. RULE.-Divide the solidity by the product of 7858, and the. square of the diameter, and the quotient, being multiplied by 3, will give the altitude. 1. The solidity of a cone is 30 feet, and the diameter 2 feet; what is the altitude? Thus, 2 x2=4; 7854 x4=3.1416; 303-1416-9.5492 × 328.6476 feet. Ans. 2. The solidity of a cone is 2513-28 feet, and the diameter 20 feet; what is the altitude? Ans. 24 feet. PROBLEM 15. To find the surface of a regular pyramid. RULE.-Multiply the perimeter of the base by half the slant height, and the product will be the convex surface; to this add the area of the base, if the entire surface is required. 1. In the regular pentagonal pyramid SA B C D E, the slant height S F is equal to 45, and each side of the base is 15 feet; required the convex surface, and also the entire surface. Thus, 15 x 575 perimeter of the base; 75 × 22 (45) = 1687.5 1687.5 square = area of convex surface. ft. And 152 225; then 225 × 1·7204774 387-107415 = the area of the base. See Table, Prob. 25, Men. of Super. Hence convex surface, Area of the base, Entire surface, 1687.5 387-107415 D Pyramid. Ans. 2074 607415 square feet. 2. What is the entire surface of a regular pyramid whose slant height is 15 feet, and the base a regular pentagon, of which each side is 25 feet? Ans. 2012-798 square feet. 3. What is the entire surface of a regular octagonal pyramid, of which each side of the base is 9.941 yards, and the slant height 15? Ans. 1073-628 square yards. 4. Required the whole surface of a triangular pyramid, each side of its base being 5 feet, and its slant height 17 feet. Ans. 157-4736 square feet, the whole surface. 5. Required the outward surface of a triangular pyramid, each side of its base being 3 feet, and its slant height 14 feet. Ans. 73.5 feet. PROBLEM 16. To find the solidity of a pyramid. RULE.-Multiply the area of the base by the altitude, and divide the product by 3; the quotient will be the solidity. 1. What is the solidity of a pyramid, the area of whose base is 215 square feet, and the altitude S O=45 feet? Thus, 215 x 45 feet. Ans. 96753: =3225 solid 2. How many solid yards are there in a triangular pyramid whose altitude is 90 feet, and each side of its base 3 yards? Ans. 38.97114 yards. 3. What is the solidity of a regular pyramid, its altitude being 12 feet, and each side of its base 2 feet? Ans. 27.5276 solid feet. Pyramid. 4. Required the solidity of a triangular pyramid whose height is 30 feet, and each side of the base 3 feet. Ans. 38.97117 feet. 5. Required the solidity of a square pyramid, each side of whose base is 30, and perpendicular keight 20. PROBLEM 17. Ans. 6000 To find the convex surface of the frustum of a pyramid. RULE.-Multiply the sum of the perimeters of the two bases by the slant height of the frustum, and the product will be the convex surface. 1. In the frustum of the regular pentagonal pyramid, each side of the lower base is 30, and each side of the upper base is 20 feet, and the slant height g H is equal to 15 feet; what is the convex surface of the frustum? Thus, 30 x 5150; 20 x 5 = 100+ 150-250÷÷2=125×15-1875 square feet. Ans. H Pyramid. 2. What is the convex surface of the frustum of a heptagonal pyramid whose slant height is 55 feet, each side of the lower base 8 feet, and each side of the upper base 4 feet? Ans. 2310 square feet. PROBLEM 18. To find the solidity of the frustum of a pyramid. RULE. Add together the areas of the two bases of the frustum of a geometrical mean proportional between them; and then multiply the sum by the altitude, and take one-third of the product for the solidity. 1. What is the solidity of the frustum of a pentagonal pyramid, the area of the lower base being 16 feet, and of the upper base 9 square feet, and altitude 7 feet? Thus, 16 x 9 = 144; 2/144 12 the mean, area of the lower base = 37 x7 = 259 ÷ 3 = 2. What is the content of a regular hexagonal frustum whose height is 6 feet, the side of the greater end 18 inches, and of the less end 12 inches? Ans. 24.681724 cubic feet. 3. How many cubic feet in a square piece of timber, the areas of the two ends being 504 and 372 inches, and the length 31 feet? Ans. 95-447. 4. What is the solidity of the frustum of a square pyramid, one side of the greater end being 18 inches, that of the less end 15 inches, and the height 60 inches? inches, solidity. Ans. 3 5. The height of the frustum of a square pyramid is 8 feet each side, the base 16 inches, and the top 10 inches; required the solidity of the frustum. Ans. 9.55 cubic feet. PROBLEM 19. To find the solidity of a wedge. RULE. Add the length of the edge to twice the length of the base, and multiply the sum by the height of the wedge, and that product by the breadth of the base, and of the last product will be the solidity. 1. The length and breadth of the base of a wedge are 35 and 15 inches, and the length of the edge is 55 inches; what is the solidity, supposing the height to be 17-14508 inches? 17-14508 x 15 x 125 6 |