Area of an Inner Rectangle

Date: 01/19/2003 at 17:00:43
From: Jamie
Subject: Word problem...

When Fred and Wilma reached retirement, they were finally able to buy 
the house of their dreams. It had a 40 ft by 72 ft rectangular yard. 
Wilma cultivated 5/12 of the area in flowers, which she grew in an 
even border around the central grass area. Every morning Fred walked 
Dino around the central grass area inside the flowers. How many laps 
did he have to do to walk at least 1/4 mile? 

I don't understand where to start because I can't visualize what facts 
are essential in arriving the answer. Is it really important that 
Wilma cultivated 5/12 of the yard in flowers? Or should I begin with 
how many feet are in a mile? 

How would you even determine 5/12 of a 40 ft by 72 ft rectangular 
yard? Is the central grass area that Fred walks Dino around the entire 
yard? Or just the remaining fraction? Please help!

Date: 01/20/2003 at 13:17:12
From: Doctor Ian
Subject: Re: Word problem...

Hi Jamie,

It's critical the the border be _even_, i.e., have the same width
everywhere:

             72
   - +---------------------+ 
  x| |                     |
   - |  +---------------+  |
     |  |               |  | 40
     |  |               |  |
   - |  +---------------+  |
  x| |                     |
   - +---------------------+
     |--|               |--|
       x                 x

Now, what is the area of the border?  We can divide it up into four
rectangles:

     +--+------------------+ 
     |  |      a        |  |
     |  +---------------+  |
     |  |               |b | 40
     | d|               |  |
     |  +---------------+  |
     |  |      c        |  |
     +---------------------+

The areas of the rectangles a-d will be functions of x.  For example, 

     area(a) = (72 - 2x) * x

Add them up, and they have to add up to 5/12 of the area of the whole
yard:

     areas of a, b, c, d = (5/12) * (72 * 40)

Solve this equation to find x, and that will tell you the dimensions
of the inner rectangle.  

Is this enough to get started? 

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