plane parallel to its base, the whole cone and the part cut off are similar; and the whole cone is to the part cut off as the cube of the diameter of the whole cone is to the cube of the diameter of the part cut off; or as the cube of the altitude of the whole cone is to the cube of the altitude of the part cut off. The volumes of spheres are to one another as the cubes of their diameters. Ex. I. The three edges of a parallelopipedon are as 3, 4, 7, and the solidity is 777924; find the edges. Here 3 × 4×7 = 84, the solidity of the parallelopipedon whose edges are 3, 4, 7; then Ex. 2. The diameter of the base of a cone is 4 ft. 8 in., and the altitude 4 ft. 6 in.; find the altitude of a portion which contains 12 cubic feet. The solidity of the cone is 253 cubic feet. .. 25 12 (4): 42·6038961; I. The three edges of a parallelopipedon are as 2, 5, 7, and the content is 259215 cubic feet; find the dimensions. Ans. 63, 163, 231, feet. 2. The diameter of a cylinder is 2 feet, and the length 3 feet; find the dimensions of a similar cylinder which shall contain 60 gallons. Ans. 2 ft. 7 in., 2 ft. 2 in. 3. The height of a cone is 7 ft. 5 in., and the diameter of the base 4 ft. 9 in.; find the dimensions of a similar cone which contains 15 cubic feet. Ans. 5 ft. 2 in., 3 ft. 3 in. 4. The transverse diameter of a prolate spheroid is to the conjugate as 7 to 4, and the solidity is 198; find the diameters. Ans. 10, 6. MISCELLANEOUS EXAMPLES. I. Find the solidity of a cube whose edge is 6 ft. 3 in. Ans. 244.1.8.3 feet. 2. The three contiguous edges of a parallelopipedon are 2, 4, 6 feet; find the solidity. Ans. 48 feet. 3. Find the length of a piece which contains 2 solid feet in the preceding question, cut off by a plain parallel to the face whose sides are 2 and 4. 4. Ans. 3 inches. Find the solidity of a cylinder whose diameter is 2 ft. 3 in., and length 3 feet. 5. Find the curve surface of a cone whose diameter is 3 feet, and slant side 4 feet. 6 in. Ans. 189 feet. 6. Find the surface of a sphere whose diameter is 6 ft. 7. Ans. 13211 feet. Find the solidity of a cone whose diameter is 4 ft. 9 in., and height 7 ft. 3 in. 8. The diameter of a sphere is 8 feet; find the curve surface of a segment whose height is 3 ft. 9 in. Ans. 94 feet. 9. Find the curve surface of the frustum of a cone whose top and bottom diameters are 4 and 6 feet, and slant side 8 feet. Ans. 125 feet. 10. Find the solidity of a sphere whose diameter is 83 inches. Ans. 350-911 feet. II. Find the curve surface of a cylinder whose diameter is 8 ft. 3 in., and length 6 ft. 8 in. Ans. 172 feet. 12. Find the number of gallons in a bucket whose top and bottom diameters are 14 and 9 inches, and the depth 12 inches. Ans. 4. 13. Find the solidity of a parallelopipedon whose dimensions are 6 ft. 8 in., 5 ft. 4 in., 8 ft. 9 in. 14. Ans. 311 feet. Find the curve surface of a cone whose diameter is 6 ft. 3 in., and slant side 6 ft. 5 in. Ans. 63 feet. 15. Find the number of gallons contained in a cylinder whose diameter is 2 ft. 3 in., and depth 3 feet. Ans. 74% 16. Find the surface of a sphere whose diameter is 18 inches. Ans. 1018. 17. The breadth of a beam is 15 inches, and the thickness 9 inches; find what length must be cut off to contain 10 cubic feet. Ans. 10 ft. 8 in. 18. Find the curve surface of the frustum of a cone whose top and bottom diameters are 6 ft. 8 in. and 8 ft. 4 in., and slant side 10 ft. 6 in. Ans. 247 feet. 19. Find the curve surface of a segment of a sphere whose height is 3 feet, and radius of base 5 feet. T. M. Ans. 106 feet. 7 20. Find the number of gallons contained in a spherical boiler whose depth is 4 feet, and radius of brim 6 feet. 21, Ans. 1620. Find the solidity of a pyramid whose height is 30 feet, the base being an equilateral triangle whose side is IO feet. Ans. 433 feet. 22. The diameter of a well is 6 feet, and the depth 18 yards; the sinking of which cost 6d. per foot for the first 6 yards, Is. per foot for the next 6 yards, and 1s. 6d. per foot for the last 6 yards. Find the cost of sinking. 23. Ans. £76. 7s. 5d. Find the content of a tapering tree whose length is 11 feet, and quarter girt 6 inches. Ans. 23 feet. 24. Find the curve surface of a cone whose height is 14 feet, and diameter of base 8 feet. Ans. 18304 feet. 25. Find the number of gallons in a bucket whose top and bottom diameters are 15 and 10 inches, and depth 13 inches. Ans. 5.832. 26. The diameter of a globe is 15 inches; find the solidity of a segment whose height is 7 inches. Ans. 795.6. 27. Find the number of gallons in a trough whose dimensions are 4 ft. 6 in., 3 ft. 9 in., 2 feet. Ans. 210. 28. How many gallons are contained in a cylinder whose diameter is 3 feet, and depth 2 feet? Ans. 88. 29. Find the curve surface of a cone whose diameter is 15 inches, and slant side 24 inches. Ans. 565 30. Find the solidity of the frustum of a cone whose top and bottom diameters are 6 and 8 inches, and depth 94 inches. Ans. 411 161 672° 31. The length and breadth of the thick end of a wedge are 20 and 6 feet, the length of the thin end 22 feet, and the perpendicular from the thin end upon the thick end 16 feet; find the solidity. Ans. 3620 yards. 32. Find the solidity of a spherical segment whose height is 2 ft. 6 in., and radius 3 ft. Ans. 43'541 feet. 33. Find the curve surface of a cone whose altitude is 12 inches, and diameter 10 inches, Ans. 2043 34. The diameter of a globe is 6 inches; find the solidity and curve surface of a segment whose height is 2 inches. Ans. 29, 37. 35. Find the weight of a cast-iron pipe whose length is 9 feet, the bore 3 inches, and the thickness of the metal I inch. Ans. 3 cwt. 33 lbs. 36. Find the surface of a sphere whose diameter is 33 inches. Ans. 23 feet. 37. Find the curve surface of the frustum of a cone whose top and bottom diameters are 8 ft. 6 in. and 19 ft. To in., and height 23 ft. 9 in. Ans. 1087. I 5 feet. 38. Find the weight of a cannon ball whose diameter is 51 inches. Ans. 21332 lbs. 39. Find the weight of a spherical shell whose diameter is 13 inches, the thickness of the metal being 14 inches. Ans. 136 lbs. 40. The length and breadth of the thick end of the frustum of a wedge are 32 and 10 feet, of the thin end 19 and 1 feet, and the perpendicular from the thin end upon the thick end 24 feet; find the solidity. Ans. 133 yards. |