Shadow Game

Date: 01/28/2001 at 16:07:34
From: Allyson Brown
Subject: Shadow game

Tom and Carol are playing a shadow game.  Tom is 6 feet tall and Carol 
is 5 feet tall. If Carol stands at the "shadow top" of Tom's head, 
their two combined shadows total 15 feet.  How long is each shadow?

Date: 01/29/2001 at 14:02:17
From: Doctor Keith
Subject: Re: Shadow game

Hi Allyson,

The lengths of their shadows are proportional to their height. Here is 
a picture to help; please note it is not to scale, since I drew it 
with characters on the keyboard. 

The vertical lines represent the height of the people in feet. Tom is 
on the right at six feet. The angled lines show us where the shadow 
will fall on the ground. The horizontal lines show us the shadows' 
lengths. We know the horizontal lengths have to be proportional and 
they must total 15 feet, but we don't how long they are individually.  
This is what we must find. 

           /|
    /|    / |
   / |   /  |
  /  |  /   |
 /   | /    |
/    |/     |
----- ------

The significance of their two different heights tells us how much of 
the shadow each of them makes up. Tom makes up 6 parts and Carol makes 
up 5 parts. These parts are units of measure, and they are similar to 
feet or inches, but we don't yet know their length. 

Together, how many equal parts of the shadow do they make up?  

Let's turn this into an equation. To start, we will define a variable, 
x. We will say that x will represent the length of each of the "parts" 
we mentioned above. Thus Tom's shadow is six "parts" or 6x long. In a 
similar way, can you see how many parts are in Carol's shadow? Because 
the shadows are end to end, how can we tell what operation (+, -, *, 
etc.) to use on the their shadow lengths to get the overall length of 
15?  

Using all this, we find the equation to be:

  15 = 5x + 6x

We need to simplify and solve for x. You can do this yourself. To 
check your answer I got, x = 15/11.

We are not quite done yet. We still have to figure out how long each 
person's shadow is. At this point we know how long a part is, so we 
can find out how long a shadow five parts long is. Carol's shadow is

  5x = 5(15/11) = 75/11 = 6 9/11 feet

So how long is Tom's?  

I hope this makes sense. If you have further questions, write back.

Best wishes,

- Doctor Keith, The Math Forum
  http://mathforum.org/dr.math/