Let [tex]x,y[/tex] be the dimensions of the rectangle. Since the perimeter is 192, we know that
[tex]2(x+y)=192 \iff x+y=96 \iff y=96-x[/tex]
The area of the rectangle is the product of the dimensions. We can use the expression for [tex]y[/tex] we just found to write it as
[tex]A=xy=x(96-x)=-x^2+96x[/tex]
This is a parabola, opening downwards, so the maximum will be its vertex, whose [tex]x[/tex]-coordinate we can find using
[tex]V_x=-\dfrac{b}{2a}=-\dfrac{96}{-2}=48[/tex]
And we deduce
[tex]y=96-x=96-48=48[/tex]
We just found out a very well known theorem: given a fixed perimeter, the rectangle with the maximum area is actually the square.
