Question 774482
Q:
Find the maximum area of a rectangle whose perimeter is 18 inches. 
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A:
Let x be the length and y be the width of a rectangle.
Perimeter, P = 2x + 2y = 18
So x + y = 9 or y = 9 - x
Area, A = xy = x(9 - x)
A = {{{9x - x^2}}} 
A is a quadratic function.
A has a maximum value because the coefficient of {{{x^2}}} is negative.
To get the maximum value, we compute the vertex (h,k) where h = {{{-b/2a}}} and k = {{{(4ac - b^2)/(4a)}}}.
In {{{9x - x^2}}}, a = -1, b = 9, and c = 0.
So h = {{{-b/2a}}} = {{{-9/(2(-1))}}} = {{{9/2}}} is the length.
and k = {{{(4(-1)(0) - (9)^2)/(4(-1))}}} = {{{81/4}}} is the area.
Therefore the maximum area of a rectangle is {{{highlight(81/4)}}} square inches.