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### The Speed of Change of a Shadow

```
Date: 12/12/95 at 21:39:16
From: Anonymous
Subject: Calculus, Rates

A street lamp whose light is 5 meters from the ground is 6 meters
from a vertical brick wall.  A person 2 meters tall walks at right
angles away from the wall directly towards the street lamp at a speed
of 1.2 meters per second.  At what speed is the shadow of the top of
the person's head descending down the wall when the person is 1.5
meters from the wall.

As with most of these problems the only difficulty I have occurs
when I try to create an equation that will allow me for solve for
dx/dt.  I'm not sure how to start it but I tried using triangle laws to
get some substitutions but nothing seemed to work.

```

```
Date: 5/29/96 at 1:54:45
From: Doctor Anthony
Subject: Re: Calculus, Rates

Let x be the distance of the man from the wall, and y be the height of

Draw a horizontal line through the top of the man's head to give two
similar triangles, which then allow us to write the following ratios as
equal.

3/(6-x) = (2-y)/x

Multiply this out to give 5x + 6y - xy = 12

Differentiate with respect to time

5(dx/dt)+6(dy/dt)-x(dy/dt)-y(dx/dt)=0

Now when x=1.5, the above ratio equation gives y=1.
Also dx/dt = 1.2
Substitute these values the into the differential equation

5(1.2)+6(dy/dt)-1.5(dy/dt)-1.2 = 0
6 + 4.5(dy/dt) -1.2 = 0
4.5(dy/dt) = - 4.8
dy/dt = -4.8/4.5
= -1.06667

The negative value is correct since the shadow is moving down the
wall and y is decreasing.

-Doctor Anthony,  The Math Forum

```
Associated Topics:
High School Calculus

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