Er. The greater chord CD of a circular zone heing 48 feet, and the lesser chord AB 30 feet, their distance FG 13 feet, required the area of the zone. See the figure to Erample Problem 13. 13+=20, and 25—20=5 the height of the lesser segment. 5 X5 3)864 324 X18 2592 32.1 2 x 48=96)5832(60 75 576 720 672 480 Prob. 20. To find the diameter of a circle whose area shall be in a given proportion to the area of a circle whose diameter is known. If the area is required to be greater than the given circle, multiply the given diameter by the square root of the intended increase, and it will give the diameter of the required circle, But if the area is intended to be less than the area of the given cire cle, divide the given diameter by the square root of the intended lecrease, which will give the diameter of the given circle. Ex. What is the diameter of a circle, whose area is 9 times as much as one of 21 inches diameter ? 9=3, then 21 X3=63 inches. Prob. 21. To find the circumference of an ellipsis, the transverso and conjugate axis being given. Multiply half t'ie zum of the two axes by 3 ; to the product add one-seventh part of the sum of the two axes, and this sum will give the circumference nor cough for most practical purposes. Er. 1. What is the circumference of an ellipsis, of which the major axis AB is 24 feet, and the minor axis CD 18 feet ? Er. 2. The width of an elliptical vault being 21 f. 7i. and the height 7f. 3ļi, what is the circumference ? Prob, 22. To find the area of an ellipsis, the major and minor ases being given. Multiply the two axes together, and the product by 7854, will give the area required. Er. What is the area of an ellipsis of which the major axis is 24 feet, and the minor 18 feet Prob. 23. To find the area of a parabola, the base or double ordinate being given, and the axis or height. Multiply the base by the height, and two-thirds of this product will be the area required. Ex. What is the area of a parabola, the axis CD beirg 12, and the double ordinai AB 18> 24. To find the area ABCD, of the frustum of a parabola, of which the parallel ends AB and CD are given , also their distance. To the square of the greatest end, add the square of the less to the product of the ends : divide the sum by the sum of the ends, mul. tiply the quotient by their distance, and two-thirds of the last product will be the answer. Er. Suppose the greater end DC 24, the less end AB 20, and their distance EF 5, required the area ABCD. 24 20 24 x 24 x 20 x 20 480 480 Def. As a line can only be measured by a line, and a surface by a surface, so a solid can only be measured by a solid ; therefore solid measure is the finding the number of cubic inches, feet, yards, &c. contained in any thing that consists in length, breadth, and thicknese. Prob. 25. To find the area of a prism. Multiply the area of the base or end, by the perpendicular height, and the product will give the solidity. Er. What is the solidity of a cube whose side is 12 inches ? 12 x 12=144; the area of the end, or 144 inches at one inch deep. 144 X 12=1728, the number of cubic inches in a cubic foot. Ex. 2. What is the solidity of a parallelopiped, whose length is 10 feet the breadtb 5f. gi. and the depth 3f. 6i. ? f. i. 5 9 3 61 Er. 3. Required the area of a triangular prism, the height being 12. 3in. one side of the base 3f. 6in. and the perpendicular of the trtangle to that side 2f. 4in. Er. 4. What is the solidity of a cylinder, whose height is 12 feet, and the diameter of the base 2:5 feet ? (2:5)ox •7854 x 12=58.905000 the solidity required. The work at length. 2.5 2 5 125 6 25 7805 12500 3125 5000 4375 4.908750 area of the base, 12 58.905000 |