2. The versed sine C E is 32, and the diameter C D 160 inches; what is the area of the segment AC B, in ale and wine gallons? Ans. 10.151 ale gallons, and 12.392 wine gallons. 3. What is the area of the segment A DB, greater than a semi-circle, in ale and wine gallons, and malt bushels; the versed sine ED being 40, and the diameter CD 62.5 inches? Ans. 7.353 ale gallons, 8.976 wine gallons, and .9642 malt bushels. PROBLEM XVI. To find the area of an ellipse. RULE. By the Pen. Divide the product of the two diameters by 359.05, 294.12, and 2738; and the respective quotients will be the area of the ellipse, in ale and wine gallons, and malt bushels. Note 1. If the two diameters of an ellipse and the number .7854 be multipled continually together, the last product will be the area of the ellipse in square inches. 2. If the sum of the two diameters of an ellipse be multiplied by 1.5708, the product will be the circumference of the ellipse, nearly. EXAMPLES. 1. Required the area of the ellipse A B C D, in ale and wine gallons, and malt bushels; the transverse diameter AB measuring 102.5, and the conjugate CD 75.2 inches. As the circular divisor on A, is to the transverse diameter on B; so is the conjugate diameter on A, to the area on B. As On A. On B. On A. On B. $21.4 359.05 2. Suppose the foregoing figure to represent the base of an elliptical cooler; what is the content in ale gallons, when the depth of the liquor is 10.8 inches? Ans. 231.8436 ale gallons. 3. The transverse diameter of an elliptical wine-vat measures 85.9, and the conjugate 63.8 inches; what is its area in wine gallons? Ans. 18.633 wine gallons. 4. A maltster has an elliptical cistern whose transverse diameter measures 96.8, and conjugate 73.2 inches; required the area and content in malt bushels, when the depth of the grain is 34.7 inches. Ans. The area is 2.587, and the content 89.7689 malt bushels. 5. The transverse diameter of an elliptical guile-tun measures 96.4, and the conjugate 82.3 inches; required the area and content, in ale gallons, when the depth of the liquor is 53.8 inches. Ans. The area is 22.096, and the content 1188.7648 ale gallons. PROBLEM XVII. To find the area of an elliptical segment, the base of which is parallel to either of the diameters of the ellipse. RULE. By the Pen. Divide the height of the segment by that diameter of the ellipse of which it is a part, and find the Area Seg. answering to the quotient, in the Table at the end of Part IV. Multiply the two diameters of the ellipse and the Area Seg. thus found, continually together, and the last product will be the area in square inches. Divide this area by 282, 231, and 2150.42, and the respective quotients will be the area in ale and wine gallons, and malt bushels. Note. If the segment be greater than a semi-ellipse, find the area of the remaining segment, which subtract from the area of the whole ellipse; and the remainder will be the area of the segment required. Or, proceed as directed in Problem XV., Rule II., Note II. EXAMPLES. 1. Required the area, in ale gallons, of the elliptical segment AC B, cut off by the double ordinate A B; the height GC being 36.4, the diameter CD of the whole ellipse 120.8, and the diameter E F 75.4 inches. inches; responding Area Seg. is .199085; then .199085 × 120.8 × 75.4=1813.3298, the area of the segment in square 1813.3298 and = 6.4302, the area in ale gallons. 282 2. What is the area of the segment AD B, in wine gallons; the dimensions being the same as in the last example? Ans. 23.1181 wine gallons. 3. Required the area in malt bushels, of the greater segment of an ellipse made by a double ordinate parallel to the conjugate diameter, at the distance of 27 inches from the centre of the ellipse; the diameters being 180 and 120 inches. Ans. 5.4282 malt bushels. PROBLEM XVIII. To find the area of a parabola, its base and height being given. RULE. By the Pen. Multiply the base by the height, and of the product will be the area in square inches, which being divided by 282, 231, and 2150.42, will give the area in ale and wine gallons, and malt bushels. Note. A parabola is a figure or section formed by cutting a cone by a plane parallel to one of its slant sides; and several parabolas may be cut from the same cone, which will all vary in their bases and altitudes, according to the distance of the cutting plane from the parallel side of the cone. (See the definitions in Part V.) EXAMPLES. 1. What is the area, in ale gallons, of the parabola ABC, whose height D B is 54.5, and the base or double ordinate A C 72.8 inches? 7935.2 3 2645.0666, the area in square inches; and 2645.06662829.3796, the area in ale gallons. By the Sliding Rule. As the square divisor on A, is to the perpendicular on B; so is of the base on A, to the area on B. 2. The absciss or height BD is 36.2, and the double ordinate AC 108.6; what is the area of the parabola Ans. 11.3458 wine gallons. ABC, in wine gallons? |