of its base, add the square of its height; then multiply the sum by the height, and the product by .5236, for the con tents. Ex. 1. What is the solidity of the segment ABD, the height ED being 4 feet, and the diameter of the base AB being 14 feet ? OPERATION. First, 7x3+4=147+16=163 Then, 163×4×.5236=341.3872 solid feet. Ex. 2. If the height of a spherical segment be 8 feet, and the diameter of its base 25 feet, what is the solidity ? Ans. 2231.5832. Ex. 3. Required the solidity of a spherical segment whose height is 6 and the radius of its base 12? Ans. 1470.27. ART. 61. The solidity of a spherical segment is frequently required when the radius of its base is not given, but if the diameter of the sphere and the height of the segment be known, the solidity may be easily found by the following Rule. From three times the diameter of the sphere, subtract twice the height of the segment; then multiply the remainder by the square of the height and the product by the decimal .5236. Ex. 1. What is the solidity of a spherical segment, whose height is 2 feet, cut from a sphere known to be 8 feet in diameter ? Then, 80x.5236=41.8880, Ans. Ex. 2. What is the solidity of a spherical segment, whose height is 3 feet, cut from a sphere whose diameter Ans. 226.1952. is 18 feet? PROBLEM XI. To find the Solidity of a Spheroid. ART. 62. Rule. Multiply the square of the revolving axis by the fixed axis; and the product multiplied by .5236 will give the solidity. Ex. 1. What is the solidity of an oblong spheroid, whose longer axis is 30, and the shorter, 20, the revolving axis being the shorter ? NOTE. If the generating ellipse revolves about its major axis, the spheroid is prolate or oblong; if about its minor axis, the spheroid is oblate. -2 OPERATION. 20×30=12000 Then, 12000x.5236=6283.2000 Ex. 1. What is the solidity of a prolate spheroid whose fixed axis is 100, and revolving axis 6 feet? Ans. 1884.96. Ex. 2. What is the solidity of an oblate spheroid whose Ans. 2094.4. axes are 20 and 10? Ex. 3. What is the solidity of an oblate spheroid, whose axes are 36 and 28? PROBLEM XII. Ans. 19010.3968. To find the Convex Surface of a Cylindrical Ring. ART. 63. Rule. To the thickness of the ring, add the inner diameter; then multiply this sum by the thickness, and the product by 9.8696 (which is the square of 3.1416) and it will give the convex surface required. Ex. 1. The thickness Ac of a cу lyndrical ring is 4 inches, and the inner diameter cd is 14 inches; re quired.the convex surface. OPERATION. Ac+cd=4+14=18 C A Then, 18×4×9.8696=710.612 sq. in. convex surface. Ex. 2. The thickness of a cylindrical ring is 3 inches, and the inner diameter 12 inches, what is the convex surface? Ans. 444.132 sq. inches. Ex. 3. The thickness of a cylindrical ring is 6 inches, and the inner diameter 20 inches, required the convex surface? Ans. 1539.6576. sq. inches. PROBLEM XIII. To find the Solidity of a Cylindrical Ring. ART. 64. Rule. To the thickness of the ring, add the inner diameter; then multiply the sum by the square of the thickness, and the product by 2.4674 (which is of the square of 3.1416) and it will give the solidity. Ex. 1. Required the solidity of an anchor ring, whose inner diameter is 8 inches, and thickness in metal 3 inches. First, 3+8=11 OPERATION. 9=sq. of thickness. Ex. 2. The inner diameter of a cylindrical ring, is 14 inches, and the thickness in metal 4 inches, what is the solidity of the ring? Ans. 710.612 solid inches. Ex. 3. Required the solidity of a cylindrical ring, whose thickness is 8 inches, and inner diameter 221 inches ? Ans. Ex. 4. What is the solidity of a cylindrical ring, whose thickness is 4 inches, and inner diameter 5 inches? Ex. 5. What is the solidity of a cylindrical ring, whose thickness is 1 inches, and inner diameter 15 inches ? Ans. PROMISCUOUS EXAMPLES. 1. What is the solidity of the greatest square prism which can be cut from a cylindrical stick of timber 2 feet 6 inches diameter, and 56 feet long? Ans. 175 cubic ft. 2. How much water can be put into a cubical vessel three feet deep, which has been previously filled with cannon balls of the same size, 2. 4. 6 or 9 inches in diameter, regularly arranged in tiers one directly above another ? Ans. 96 wine gallons. 3. What will be the expense of painting a conical spire, at 8 cents per square yard, whose height is 120 feet and circumference at the base 60 feet? Ans. $32. 4. What is the solidity of a conic frustrum whose height is 40 feet, the greater diameter 18 feet, and the smaller, 9 feet? Ans. 5. What is the solidity of the greatest cube which can be cut from a sphere three feet in diameter ? Ans. 5 ft. 6. What is the solidity of a cylinder whose height is 20 feet, and the circumference of the base 20 feet ? Ans. 636.64 cubic ft. 7. What is the solidity of a spherical segment whose height is 26 feet, and the diameter of the base 48 feet? Ans. 8. How many such globes as the earth are equal in bulk to the sun, if the former is 7930 miles in diameter, and the latter 890,000 ? Ans. 1.413.678. 9. The diameter of a sphere is 12 feet; required its solidity ? Ans. 10. What is the solidity of a cylindrical ring whose thickness is 8 inches, and inner diameter 19 inches ? 11. What is the solidity of a prolate spheroid, whose axes are 55 and 33? Ans. 31361.022. 12. What is the solidity of an oblate spheroid whose axes are 67 and 48 feet? Ans. |