2. What is the area of a circle of which the radius is 3 feet? 3. What is the area of a circle of which the diameter is 24 feet? Ans., 452 3904 square feet. 4. What is the area of a circular plot of grass of which the radius is 5 feet? Ans., 78*54 square feet. 5. What is the circumference of a circle of which the radius is 3 feet? Ans., 18 8496 feet. 6. What is the circumference of a circle of which the diameter is 6 feet? Ans., 18 8486 feet. 7. What is the area of a circle, the radius being 3 inches? Ans., 28 27446 square inches. 8. What is the area enclosed between two concentric circles, the diameters of which are respectively 20 and 15 feet? Ans., 137 445 feet. 9. Find the above when the radii are 3 and 2 feet Ans., 15 708 feet. 10. Find the above when the diameters are 16 and 10 feet respectively. Ans., 122 5224 square feet. respectively. 11. Find the area as above when the diameters are 21.75 and 9.5 feet. Ans., 300 6609 square feet. CUBIC OR SOLID MEASURE. I. To find the solidity of a cube. RULE.-Multiply the length of the side of the cube by itself, and the product by the same quantity; or Solidity 73 where l is the length of the side. Example. Find the solid content of a cube which measures 6 inches in the side. By Fractions. Content = 2×6×6/ 13 x 13 x 13 160 × 13 2197 1. Find the solid content of a cubical block of marble, the length of the side of which is 17 feet. Ans., 4913 cubic feet. 2. What is the content of a cistern, the length breadth, and depth of which equally measure 9 feet? Ans., 729 cubic feet. 3. What is the quantity of earth dug out of an excavation 6 feet deep, 6 feet long, and 6 feet wide? Ans., 216 cubic feet. II. To find the solid content of a rectangular block. RULE.-Multiply the length by the breadth. and the product by the depth; the ultimate product is the solid content required. Example.-How much earth is dug out of a trench which is 17 feet long, 3 feet deep, and 12 feet wide ? Solid content = 17x3x 12=612 cubic feet. EXERCISES. 1. Find the solid content of a rectangular block of marble which is 9 feet long, 3 feet broad, and 6 feet deep. Ans., 189 cubic feet. 2. What is the content of a cistern which measures 3, 71, and 94 feet respectively, in breadth, depth, and length ? Ans., 208 cubic feet. 3. If the cubic content of a block of marble be 4913 feet, what is the length of the side? Ans., 17 ft. 4. If the cubic content of a mass of earth be 729 feet, what is the surface of one side or face of it? Ans., 81 square feet. 5. The solid content of a block of iron is 208 cubic feet; the length is 94 feet, and the breadth 7 feet; what is the depth? Ans., 3 feet. 6. The cubic content of a cistern is 189 feet; the length is 9 feet, and the breadth 6 feet; what is the depth? Ans., 3 feet. 7. How many blocks, each 3 inches long, 7 inches broad, and 2 inches deep, can be cut out of a rectangular solid block, measuring 3 feet long, 7 ft. broad, and 2 feet deep? Ans., 1728. III. To find the solid content of a cylinder. This will evidently depend upon two factors:— 1. The sectional area. 2. The length of the cylinder. Now the sectional area is that of a circle, and is found by the rule already given; therefore the solid content will be the product of the two factors, or the area of base multiplied by the length of the cylinder. Example. Find the solid content of a cylinder, of which the diameter is 6 feet and the length 16 feet. Surface of base = 62 x 7854 .. solidity = 62x·7854 × 6 =216 x 7854 = 169 6464 cubic feet. EXERCISES. 1. A cylindrical rod 11 feet long has an area in transverse vertical section of 3 square feet; what is the solid content of the cylinder? Ans., 33 cub. ft. 2. The radius of a cylinder is 4 inches, and the length 17 feet; what is its solid content? Ans., 10254 1824 cubic inches. 3. If the cubic content of a cylinder be 169.6464 cubic feet, what is the surface of the base if the length be 12 feet? Ans., 14 1372. 4. If the cubic content of a solid cylinder be 33 cubic feet, and the area of the base 3 square feet, what is the length of the cylinder? Ans., 11 feet. 5. What is the solid content of a cylinder of which the radius of the base is 10 feet and the length of the cylinder 10 feet? Ans., 3141 6 feet. 6. What is the solid content of a cylinder of which the base is 6 feet in diameter and the length 6 feet? Ans., 169 6464 cubic feet. 7. What is the solidity of a cylinder 10 feet long and of which the diameter is 24 feet? Ans. 4523.904 ft. 8. What is the solidity of a cylinder of which the diameter is 10 feet and the height 7 feet? Ans., 549 78 feet. 9. What is the solidity of a cylinder 12 feet long, of which the radius is 3 feet? Ans. 339 2928. 10. What is the solidity of a cylinder 3 feet long, if the circumference of the base be 3 feet? Ans., 2.14+8. 11. The solidity of a cylinder is 18 8496 cubic inches, the length is 1 inch; what is the diameter of the base? Ans., 24. 12. What is the solidity of a cylinder of which the height is 1 foot and the diameter of the base 6 feet? Ans., 28 27446 cubic feet. IV. To find the solid content of a triangular vertical prism. As in the preceding instance the solidity depends on two factors: 1. The area of the triangular base. 2. The length of the prism. The solidity of the prism will therefore be the product of these two factors; or, Solidity area of base x length. Now the area of the base may be determined, 1. When the base and perpendicular height are given, since Area of base = And solidity = Perpendicular x base 2 Perp. base × length 2 And (2), when the three sides of the base are given, since Areas (-a)(s—b)(s—c) × and soliditys (s—a)(s—b)(s—c) x length. Example. Find the solidity of a triangular vertical prism, of which one side of the base is 6 feet, the perpendicular let fall upon it is 2 feet, and the length of the prism 12 feet. Solidity = "x2× 12 = 72 cubic feet. 2 Find the solid content of a vertical triangular prism of which the three sides containing the triangular face are 6 inches respectively, and the length 18 in. Area of base = √9×3×3×3 Solidity 93 x 18 = 162/3. EXERCISES.-1. Find the solid content of a triangular vertical prism 12 inches long, of which the sides of the triangular base are 24, 12, and 18 inches. Area of triangular base = √s(—a)(s—b)(s—c)=27/15. Solidity 27/15 x 12 x 324/15. MISCELLANEOUS EXERCISES. 1. What will the paving of a court cost at 4 d. per yard, the length being 58 ft. 6 in. and the breadth 54 feet 9 inches? Area = 58 × 54% = 117 x 219 117 × 219 8 2 2=32023 sq. ft. 355 sq. yds. = Cost =3553x42=169031d. = 140s. 1031d. = £7 16s. 1031. 2. A block of stone is 4 ft. long, 24 ft. broad, and 1 ft. thick; it weighs 27 cwt.; find the weight of 100 cubic inches of the stone. Content = 4×2×11=4××2=25-12 cub. ft. =12x1728 cubic inches = 21600 cubic inches. 21600 100 :: 27 cwt.: x. x= cwt. = = 3. How many feet are there in 1795 miles? 1795 x 5280 feet 9477600 feet. 4. Find by practice cost of 36 mls. 3 fur. 22 yards of telegraph wire, at £14 13s. 4d. per mile. |