Differentiation ProblemDate: 11/15/97 at 21:37:53 From: Carolyn Confer Subject: Calculus: differentiation problem A light shines from the top of a pole 50 ft. high. A ball is dropped from the same height at a point 30ft away from the light. How fast is the shadow of the ball moving along the ground 1/2 second later? (Assume the ball falls s = 16t^2 ft.in in t seconds). I have spent a few hours on this problem this weekend, drawing all sorts of diagrams, finding angles, etc. But I cannot find the formula for the speed of the shadow. Any help/advice would be greatly appreciated. Date: 11/16/97 at 08:12:14 From: Doctor Anthony Subject: Re: Calculus: differentiation problem Draw a figure to represent the situation when the ball has fallen a distance s feet from its initial position. Let x = distance along the ground from the point immediately below the ball to the point where the shadow of the ball meets the ground. We then have two similar triangles and can write down the ratio of corresponding sides in the form: x 30 ------- = ------- 50-s s We also have ds/dt = 32t and we are required to find dx/dt when t = 1/2 Multiplying out we get xs = 1500 - 30s and differentiating x(ds/dt) + s(dx/dt) = -30(ds/dt) at t=1/2 ds/dt = 16 s = 16(1/4) = 4 and 4x = 1500 - 120 x = 345 ft So 345(16) + 4(dx/dt) = -30(16) 4(dx/dt) = -6000 dx/dt = -1500 ft/sec So at this moment the shadow is moving at a speed 1500 ft/sec towards the point immediately below the ball. The high speed is to be expected since initially the shadow is at infinity and has to move to the end point in 1.76 seconds. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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