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Why a Square Maximizes Area/Perimeter

Date: 07/24/2002 at 10:21:44
From: Elizabeth
Subject: area and perimeter

I need to come up with a rectangle with an area greater than 16 ft. 
squared, and a perimeter of 16 ft. 


Date: 07/24/2002 at 12:27:34
From: Doctor Ian
Subject: Re: area and perimeter

Hi Elizabeth,

Has someone led you to believe that this would be possible?  A square 
with perimeter 16 ft has an area of 16 ft^2:

    4
  +---+
  |   | 4    
  +---+

Suppose we keep the same perimeter, but use another shape.  To do 
this, we would have to reduce the width by some amount, x, and 
increase the length by the same amount:

   4-x
  +--+
  |  |
  |  | 4+x   
  +--+

Now the perimeter is still 16 feet, but the area is 

  A = (4+x)(4-x)

    = 4^2 - x^2

    = 16 - x^2

square feet.  Which is to say, _any_ change we make, from a square to 
a non-square rectangle, will result in a smaller area. 

Does that make sense?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Triangles and Other Polygons

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