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!
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