(2) Mr. Wilson is standing near the top of a ladder 24 feet long which is leanin

Question

(2) Mr. Wilson is standing near the top of a ladder 24 feet long which is leaning against a vertical wall of his house. Dennis, the little boy next door, ties a rope from his tricycle to the bottom of the ladder and starts to pull the foot of the ladder away from the house wall. The bottom end of the ladder begins to slide away from the wall at the rate of 1 foot per second. (a) How fast is the top of the ladder sliding down the wall when the foot of the ladder is 8 feet from the wall? [2 points] (b) How fast is the top of the ladder sliding down the wall when the top of the ladder is 16 feet from the ground? [2 points] (c) How fast is the angle between the top of the ladder and the wall changing when the foot of the ladder is 8 feet from the wall? [2 points]

Explanation / Answer

let x be horizontal dispacement of bottom

y be vertical displacement of top of ladder

x2 + y2 = 576

differentiating w.r.t t we get 2x dx/dt + 2y dy/dt =0

(a)at x= 8 feet and dx/dt=1

y= 242-82 = 22.62 feet

dy/dt = -x/y dx/dt = -8/22.62 = 0.35366 feet/s downward

(b)

at y = 16 feet x/dt = 1 ; x = 576-256 = 17.88 feet

dy/dt = -17.88/16 = 1.12 feet/s downward

(c)

tan = x/y

sec2 d/dt = yx'-xy'/y2

d/dt = cos2 [(22.62 - 8*.35366)/22.622] = 0.034 rad/s

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