Maximum Area of a Rectangle with Fixed Perimeter

Date: 03/03/2004 at 11:02:13
From: John
Subject: area of a rectangle

If I am given a specific length of fence, such as 128 feet, how can I 
calculate the maximum amount of square footage that I can enclose in a 
rectangle using the fence?

At first I thought that given a specific fixed length of fence any
rectangle would yield the same square footage.  But that's not true. 
I believe that there must be an equation to figure this out.

I used guess and check and ultimately found that a square of 32 x 32
feet yielded the most square footage.  Then altering it slightly to
make a rectangle was the best I could come up with.  But it was all
guessing.

Date: 03/03/2004 at 11:51:33
From: Doctor Douglas
Subject: Re: area of a rectangle

Hi John,

Thanks for writing to the Math Forum.

Actually, the 32 ft x 32 ft square gives the maximum area of any
rectangle of fixed perimeter 128 ft.  

Here's how to convince yourself of this.  Any rectangle must have a 
length L and a width W.  For the perimeter to be 128, 2L + 2W = 128, 
or L + W = 64.  The area of this rectangle is L*W.  We want to pick 
the L and W such that we get the maximum area L*W.

Now, let's rewrite our equations as follows.  Let the length be 
L = 32 + x and the width be W = 64 - L = 64 - (32 + x) = 32 - x.  Our
task now is to see what value of x gives the best (maximal) area:

   area = L*W = (32 + x)(32 - x)
              = 1024 + 32x - 32x - x^2
              = 1024 - x^2

Now, x^2 is a positive number for any x not equal to zero, so that we 
will always reduce our area from 1024 sq ft, which is achieved only at 
x = 0, or L = W = 32 ft.
  
This argument doesn't use calculus, which is a more powerful method to 
compute maximum and minimum quantities and where they occur.

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

Date: 03/03/2004 at 13:02:13
From: John
Subject: area of a rectangle

Thank you for your quick response.  I was going to point out that the
32 by 32 arrangement is a square and I am looking for the rectangle of
largest area, but then I remembered that a square is in fact also a
rectangle!  Thanks for all your help on this!