PROBLEM XX. 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 circle required. But if the area is intended to be less than the area of the given circle, divide the given diameter by the square root of the intended decrease, which will give the diameter of the given circle. EXAMPLE I. What is the diameter of a circle, whose area is 9 times as much as one of 21 inches diameter? √9-3, then 21x3=63 inches. EXAMPLE II. What is the diameter of a circle, whose area is of a circle of 21 inches diameter? √93, then =7. PROB PROBLEM XXI. To find the circumference of an ellipsis, the transverse and conjugate axis being given. Multiply half the sum of the two axes by 3; to the product add part of the sum of the two axes, and this sum will give the circumference near enough for most practical purposes. EXAMPLE 1. What is the circumference of an ellipsis, whose transverse axis is 24 feet, and the conjugate 18 feet? 24 +18 2)42 21 X3 63 +3 66 feet the circumference. EXAMPLE II. The width of an elliptical vault being 21f. 71. and the height 7f. 3i. what is the circumference? 2)21 7 10 9 6 18 1 54 3 +2 7 56 10 the circumference. ft. in. PROBLEM XXII. To find the area of an ellipsis, the transverse and conjugate axes being given. Multiply the transverse axis by the conjugate, and the product by 7854, will give the area required. EXAMPLE. What is the area of an ellipsis whose transverse axis is 24 feet, and the conjugate 18 feet? 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. VOL. I. Bb EX EXAMPLE. What is the area of a parabola, the axis CD being 12, and the double ordinate AB 18? 12 X18 3)216 72 X2 144 answer. PROBLEM XXIV. To find the area ABCD, of the frustum of a parabola, whose parallel ends AB and CD are given; also their distance EF. To the square of the greatest end, add the square of the lesser end, to the product of the ends: divide the sum by the sum of the ends, and the quotient multiplied by the distance of the ends, two thirds of the product will be the answer. |