4. What is the solidity of the frustrum of a cone, the circumference of the greater end being 40, that of the less end 20, and the length or height 50 ? Ans. 3713.7333. Let s=tabular segment, whose versed sine is Br-D, soms tab. seg. whose versed sine is Br—(D—d)--d, and the other letters as above. Br BI-D Br th Then (sXD_SX ds X ✓ 4) * -d d solidity of the elliptic hoof EFBD. 1 d2 And v 4h + (D—d) x (seg. FBE х D 4x (+d)-Ar dar X seg. of the circle ÁB, whose d-Ar height is DXã )=equal convex surface of EFBD. D В" -X da ar III. When the section is parallel to one of the sides of the frustrum. Let A = area of the base FBE, and the other letters as before A XD Then 40/(1--d)xd) th=solidity of the para. bolic hoof EFBD. 1 And v 4h2 x (D-d) x (seg. FBE Di x D VdXDd)=convex surface of EFBD. IV. When the section cuts off part of the base, and makes the angle DrB greater than the angle CAB. 5. What is the solidity of the frustrum of a square pyramid, one side of the greater end being 18 inches, that of the less end 15 inches, and the height 60 inches ? Ans. 16380 inches. 6. What is the solidity of the frustrum of an hexagonal pyramid, the side of whose greater end is 3 feet, that of the less end 2 feet, and the length 12 feet? Ans. 197.453776 feet. PROBLEM X. RULE.* Add twice the length of the base to the length of the edge, and reserve the number. D D A B F Let the area of the hyperbolic section EDF=A, and the area of the circular seg. EBF=a. dx Er Then ī* (aXD-AX -)=solidity of the hyper. bolic ungula EFBD. 1 And à x V 4h+(D—d) x (cir. seg. EBF ? —-X Br~(D-d) ✓ =curve surface of EFBD. Br-D-d Brd-D Note.—The transverse diameter of the hpy. seg. = -d and the conjugate=dvo-d-Br, from which its area may be found by the former rules. * Demon. When the length of the base is equal to half of the wedge, the wedge is evidently equal to half a prism of the same base and altitude. Br dx cr Br D -BT Multiply the height of the wedge by the breadth of the base, and this product by the reserved number; 1 of the last product will be the solidity. EXAMPLES. 1. How many solid feet are there in a wedge, whose base is 5 feet 4 inches long, and 9 inches broad, the length of the edge being 3 feet 6 inches, and the perpendicular height 2 feet 4 inches ? (64x2+42) x 28 x 9 (128+42) x 28 x 9 Here 6 6 170 x 28 x 9 170 x 28 x 3 170 x 14x3=7140 solid 6 2 inches. And 7140-1728 = 4.1319 solid feet, the content required. And according as the edge is shorter or longer than the base, the wedge is greater or less than half a prism, by a pyramid of the same height and breadth at the base with the wedge, and the length of whose base is equal to the difference of the lengths of the edge and base of the wedge. Therefore, let the length of the base BC=L; the length of the edge EF=l; the breadth of the base BA=b; and the height of the wedge EP=h; and we shall have by the former rules +1+1 bih +bhx 31+2L_21 =bh x 2 3 2 3 6 21+7 Q. E. D. blh +bhx 1 =bh X 2. The length and breadth of the base of a wedge are 35 and 15 inches, and the length of the edge is 55 inches. what is the solidity, supposing the perpendicular height to be 17.14508 inches ? Ans. 3.1006 feet. PROBLEM XI. To find the solidity of a prismoid. RULE. * To the sum of the areas of the two ends add four times the area of a section parallel to and equally distant from both ends, and this last sum multiplied by $ of the height will give the solidity. Note.-The length of the middle rectangle is equal to half the sum of the lengths of the rectangles of the two ends, and its breadth equal to half the sum of the breadths of those rectangles. * Demon. The rectangular prismoid is evidently composed of two wedges, whose heights are equal to the height of the pris. moid, and their bases its two ends. Wherefore, by the last h problem its solidity will be=(21+1xB+21+1xb)* z. L+1, , which, by putting m= and m = becomes 2 2 h BL+61 +4mm x 6; which is the rule, as was to be shown. The solidities of the two parts, commonly called the ungulas, or hoofs, into which the frustrum of a rectangular pyramid is divided, may be found by the last two rules, as they are only composed of wedges and prismoids. A very elegant demonstration of this rule for the prismoid may be een in Simpson's Fluxions, p. 179, 2d Edition. If the bases of the prismoid are dissimilar rectangles, of which L, 1 and M, m are corresponding dimensions, and h the height; Then (2+Ll.x+21+ L.m) xth=solidity. EXAMPLES. 1. What is the solidity of a rectangular prignoid, the length and breadth of one end being 14 and 12 inches, and the corresponding sides of the other 6 and 4 inches, and the perpendicular 30 feet. Here 14x 12+6X4=168+24=192=sum of the area. of the two ends. 14+6 20 Also =10=length of the middle rectangle. 2 2 12+4 16 And =8=breadth of the middle rectangle. 2 2 Whence 10 x 8X4=80 X 45320=4 times the area of the middle rectangle. 366 Or (320+192)* 6 =512x 61=31232 solid inches. And 31232--1728=18.074 solid feet, the content. 2. What is the solid content of a prismoid, whose greater end measures 12 inches by 8, the less end 8 inches by 6, and the length, or height, 60 inches ? Ans. 2.453 feet. 3. What is the capacity of a coal wagon, whose inside dimensions are as follow : at the top, the length is 81), and breadth 55 inches; at the bottom the length is 41, and the breadth 294 inches; and the perpendicular depth is 47% inches? Ans. 126340.59375 cubic inches. |