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### 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
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/
```
Associated Topics:
High School Calculus
High School Physics/Chemistry

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