. Differential and integral calculus. ittedat the rate of 10 cu. ft. a second, at what rate is the pistonmoving ? 62 Differential Calculus Let y = volume and x = altitude of cylinder at any instant;then = »£) dy = - rtfr,4 40 .*. dx — — feet a second. 7T 6. Gas is introduced into a thin elastic spherical film at therate of 10 cu. ft. a second. At what rate is the radius increas-ing when the volume is cu. ft. ? 3 Let y = volume and x variable radius ; then y = -ttxs, .. dy = 4 7rx2dx,3 dx 43dy 10 1 4 TTX2 4OO 7T 40 7T feet a second. 7. A man 6 feet in height, walking at the rate of 2 milesa
. Differential and integral calculus. ittedat the rate of 10 cu. ft. a second, at what rate is the pistonmoving ? 62 Differential Calculus Let y = volume and x = altitude of cylinder at any instant;then = »£) dy = - rtfr,4 40 .*. dx — — feet a second. 7T 6. Gas is introduced into a thin elastic spherical film at therate of 10 cu. ft. a second. At what rate is the radius increas-ing when the volume is cu. ft. ? 3 Let y = volume and x variable radius ; then y = -ttxs, .. dy = 4 7rx2dx,3 dx 43dy 10 1 4 TTX2 4OO 7T 40 7T feet a second. 7. A man 6 feet in height, walking at the rate of 2 milesan hour, passes under an electric light 18 feet above the pave-ment. Assuming the pavement to be horizontal, find (a) therate at which the mans shadow is lengthening ; (£) the velocity D of the end of his shadow; (V)the rate at which he is recedingfrom the light when 15 feetfrom its foot. Let D be the light, and BM•L the position of the man at anyFig. 6. instant. (a) Let AB = y, BC — x\ then, from similar triangles, y y + *. hence, 6 18 1 y = -*; dy = -dx = 1 mile per hour. Analytical Applications 63 {b) Let AC = y and BC = x; then y 18 y 6 X 5 .\j> = =!- = ^-dx 3 miles per hour. hence, f/y z (c) Let ^4 be the mans position, and let AD = y and ^4C = x\then, f=x2+ 324; ?# I c hence, dy = - dx = — 2 — i\ miles per hour, nearly. 8. Two ships start from Sandy Hook at the same time, onegoing N. 300 E. at the rate of 10 miles an hour, the other goingdue east at the rate of 12 miles an hour. At what rate arethey separating at the end of two hours ? Let y = distance of ships apart at the instant, and let u andv be the varying distances of the vessels from Sandy Hook;then y = u2 -f- Z-2 — 2 uv cos 6o° = u2 -f- v1 — uv; .. 2 ydy = 2 udu -f- 2 vdv — ndv — vdu; (2 u — 7) du 4- (2 v — 11) dv _ ?*? } — / „ , , =- — 2 V31 miles an hour. 2 Var+ 7^— ?/^ 9. The altitude of an equilateral triangle is 2 feet, and isincreasing at the rate of 2 inches a minute.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1918
