1. The side of a cube is 18 inches; what is the area of its surface? = = Thus, 18 x 18: 324 x 6 = 1944 in. 144 13.5 square feet. 2. The side of a cube is 25 inches; what is the area of the surface? Ans. 26 feet. 3. The side of a cube is 19 feet; what is the area of its surface?. Ans. 2166 feet. 4. The side of a cube is 25.5 feet; what is the area of its surface? Ans. 3901.5 feet. 5. The side of a cube is 36 yards; what is the area of its surface in square feet? PROBLEM 2. The area of the surface of a cube being given, to find the length of the side. RULE.-Divide the area by 6, and extract the square root of the quotient. 1. The area of a cube is 2400 square inches; required the length of the side. = 20 inches. Ans. Thus, 24006 = √400 2. The area of a cube is 216 square feet; what is the length of the side? Ans. 6 feet. 3. The area of a cube is 5400 square inches; what is the length of the side? Ans. 2 feet. 4. The area of a cube is 258 square feet; what is the length of the side? Ans. 6.55+. 5. The area of a cube is 1800 square inches; what is the length of the side? PROBLEM 3.. To find the solidity of a cube, the length of one of the sides being given. RULE.-Cube the given side. 1. The side of a cube is 25.5 inches; what is the solidity? Thus, (25-53) that is, 25.5 x 25.5 x 25.5 16581-375 cubic inches. = 2. The side of a cube is 15 inches; what is the solidity? 1.9531 feet. Ans. Ans. 216 feet. 3. The side of a cube is 6 feet; what is the solidity? PROBLEM 4. To find the side of a cube, the solidity being given. RULE.-Extract the cube root of the solidity. 1. What is the length of the side of a cube containing 36 solid feet? 3/36=3·3019, side required. 2. What is the length of the side of a cube whose solidity is 1800 inches? Ans. 12-1644 inches. 3. What is the length of the side of a cube whose solidity is 789 cubic feet? Ans. 9-2404 feet. 4. What is the length of the side of a cube whose solidity is 2984 yards? PROBLEM 5. To find the solidity of a parallelopipedon. RULE.-Multiply the length by the breadth, and that product again by the depth, or altitude, and it will give the solidity required. 1. Required the solidity of the parallelopipedon A B C D E F G H, whose length A B is 8 feet, its breadth 4.5 feet, and depth or altitude A D, 6.75 feet. G 8 Thus, A B x ADXFD: × 6·75 × 4.5 = 54 × 4.5 = 243 solid feet. Ans. Parallelopipedon. E H 2. What is the solidity of a block of marble whose length is 10 feet, breadth 5.75 feet, and depth 3-5 feet? Ans. 201-25 feet. 3. The length of a parallelopipedon is 36 inches, the width 20 inches, and the depth 18 inches; how many solid feet will it contain? Ans. 7.5 feet. 4. How many bushels are contained in a bin 5.5 feet in length, 4.75 feet in width, and 3.75 in depth? Ans. 78.724 bushels. 5. What is the solidity of a block of marble whose length is 12 feet, breadth 5, and depth 24 feet? Ans. 172 feet. 6. The length of a parallelopipedon is 15 feet, and each side of its square base 21 inches; what is the solidity? Ans. 45.9375 feet. PROBLEM 6. To find the solidity of a prism. RULE.-Multiply the area of the base into the perpendicular height of the prism, and the product will be the solidity. 1. What is the solidity of the triangular prism ABCDEF, whose length A B is 20 feet, and either of the equal sides B C, C D, or D B of one of its equilateral ends B C D, 5 feet? See Problem 25, Mensuration of Superficies. Thus, the area of the base is 5 x 5 = 25; and 433013 x 25 = 10.825325 x 20216.5065 ft. solidity required. 2. What is the solidity of a triangular prism whose length is 18 feet, and one side of the equilateral end 1.5 feet? = Prism. Thus, the area of the triangular base is 1.52 2.25 x ·433013 = 97427925 x 18 17.5370265. Ans. 3. What is the value of a prism whose height is 32 feet, and each side of the equilateral end 14 inches, at 20 cents per solid foot? Ans. D. 3.772. 4. What is the solidity of a regular pentagonal prism whose altitude is 20 feet, and each side of the base 15 feet? = 225; Thus, 152 the area of the base. and 1.7204774 × 225 387-107415 = solidity. 5. What is the number of cubic or solid feet in a regular pentagonal prism of which the altitude is 15 feet, and each side of the base 3.75 feet? Ans. 362.913. PROBLEM 7. To find the convex surface of a cylinder. RULE.-Multiply the circumference or peri- D phery of the base by the height of the cylinder, and the product will be the convex surface required; to which add the area of each, and the sum will be the whole surface of the cylinder. 1. What is the convex surface of the right cylinder A B C D, whose length B C is 24 feet, and the diameter of its base 16 feet? Thus, 3-1416 x 1650.2656, the circumference of the base. B And 50-2656 × 24 face required. = 1206-3744 square feet, the convex sur 2. What is the whole surface of a right cylinder, the diameter of whose base is 2-5 feet, and height 5 feet? Thus, 3-1416 x 2.5 7.854, the circumference of the base. 7854 = ends. Then 39-279-817549-0875 square feet, the whole surface. 3. What is the convex surface of a cylinder, the diameter of whose base is 20, and altitude 50 feet? Ans. 3141.6 square feet. 4. What is the convex surface of a cylinder whose base is 30 inches, and altitude 5 feet? Ans. 5654-88 sq. inches. 5. What is the whole surface of a right cylinder, the diameter of whose base is 16 inches, and its length 20 feet? Ans. 86.5685 feet. PROBLEM 8. To find the solidity of a cylinder. RULE.-Multiply the area of the base by the perpendicular height of the cylinder, and the product will be the solidity. 1. What is the solidity of a cylinder, the diameter of whose base is 40 feet, and altitude 25 feet? Thus, 402 = 1600 × 7854 = 1256·64 : area of the base. Then 1256.64 × 25 = 31416 solid feet. Ans. 2. What is the solidity of a cylinder whose height is 5 feet, and the diameter 2 feet? Ans. 15.708 solid feet. 3. What is the solidity of a cylinder whose altitude is 12 feet, and the diameter of its base 15 feet? Ans. 2120.58 cubic feet. 4. The length of a cylinder is 30 feet, and the diameter 20 inches; what is the solidity? Ans. 65-45 solid feet. 5. How many solid feet in a round stick of timber 16 feet long, and the diameter at each end 15 inches? Ans. 19.635 solid feet. 6. Required the solidity of a cylinder, the diameter of whose base is 30 inches, and height 50 inches? Ans. 20.4531 solid feet. PROBLEM 9. To find the whole surface of a right cone. RULE.-Multiply the circumference of the base by the slant height, or the length of the side of the cone, and half the product will be the area of the convex surface; to which add the area of the base, and the sum will be the whole surface of the cone. 1. What is the convex surface of the cone whose vertex is C, the diameter A D of its base being 8.5 feet, and the side CA, 50 feet? Thus, first 3.1416 x 8.5 = 26.7036 = circumference of base. Then 26-7036 x 50÷2-667-59, convex surface. 2. The diameter of a cone is 4.5 feet, and the slant height 20 feet; required the convex surface. Ans. 141-372. 3. The diameter of the base of a cone is 3 feet, and the slant height 15 feet; what is the convex surface? Ans. 70-686 sq. ft. Vertex. Cone. 4. The slant height of a cone is 20 feet, and diameter 3 feet; required the surface of the cone. Ans. 101-3166 square feet. 5. The circumference of the base of a cone is 10-75, and the slant height is 18-25; what is the entire surface? Ans. 107-29021 square feet. PROBLEM 10. To find the solidity of a cone. RULE.-Multiply the square of the diameter of the base by 7854, and that product by one-third of the perpendicular altitude; the product will be the solidity. 1. Required the solidity of a cone, the diameter of whose base is 18 inches, and its altitude 15 feet. Thus, 18 in. = = = 1.52 x 7854 = 1.26715 area of base. And 176715 x 15 (5) 8.8357 feet, solidity required. 2. What is the solidity of a cone, the area of whose base is 380 square feet, and altitude 48 feet? Thus, 380 x 48 = 18240 ÷ 3 = 6080 feet. Ans. 3. Required the solidity of a cone whose altitude is 10 feet, and the circumference of its base 9 feét. Ans. 22.5609 cubic feet. |