Question 1069393
{{{system(2x+2y=48,xy=A)}}}


Area A is a function of either x or y, and the perimeter equation can be simplified.


{{{system(A=xy,x+y=24)}}}


{{{y=24-x}}}
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{{{A=x(24-x)}}}
{{{highlight(A(x)=x(24-x))}}}
No need to try multiplying the quadratic expression.  This is a parabola function and it has vertex as a maximum.  


{{{x(24-x)=0}}}
Roots or zeros?
x=0 or x=24.


The maximum occurs in the exact middle between the roots.

{{{(0+24)/2=12}}}


A will have its maximum, the vertex, at x=12.


Dimensions will be 12, and 24-x=24-12=12;
meaning the shape will be a square.


Maximum area will be {{{xy=12*12=highlight(144)}}}.

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THE SHAPE FOR THE MAXIMUM AREA OF A RECTANGLE IS  A SQUARE.