Differentiation Problem

Date: 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
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