Among all rectangles that have a perimeter of 210, find the dimensions of the one whose area is largest. write your answers as fractions reduced to lowest terms.

Let the length of the rectangle be  x units then the width = (210 - 2x) / 2

= 105 - x  units

Area A = x(105 - x) = 105x - x^2

Finding  the derivative:_

dA/dx = 105 - 2x  = 0   for  max/minm volume

x = 52.5

this is for a maximum area because  second derivative  is negative ( = -2).

width = 105 - 52.5 = 52.5

Dimension for maximum area = 52 1/2 * 52 1/2.  ( The rectangle is actually a square).

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