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put this solution on YOUR website! Let te dimensions of the rectangles be x,y.
Since the perimeter is 2(x+y)= 20,so x+y = 10.
The area A = xy = x(10-x) = -x^2 + 10x
(by complete square)
= -(x^2 - 10x + 25) + 25
= -(x - 5)^2 + 25
(since -(x - 5)^2 <= 0 for all x)
A has max value if x =5 and so y = 10-5 =5.
Hence, when the rectangle dimensions are 5, 5 , its area is max.
Also, note that in this case, Area/perimeter = 25/20 = 1.25
If the shape is a circle, then we see that
Area/perimeter = (pi R^2)/(2pi R) = R/2
Therefore, when the radius R > 2.5 , the circle is more efficient
than the rectangle. (higher area to perimeter ratio)
Kenny
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