A rectangular flower bed has a width of x feet and a perimeter of 104 feet. Find x such that the area of the rectangle is maximum

Question

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Answer 1

The maximum area is when the rectangle is a square.

If x is the side of the square, then its perimeter is 4x and width=length=x. So 4x=104 and x=104/4=26 feet.

(To prove that the maximum area is when the rectangle is a square we start with a square of area x². Then we shorten the width and increase the length by an amount h. Width=x-h and length=x+h. Width+length remains unchanged at 2x, so that means the perimeter, 4x, is also unchanged. The new area is (x+h)(x-h)=x²-h². The maximum value for this expression is x², when h=0. So length=width=x, a square.)