Question 817049
Let x = a dimension and y = the other dimension.

A is for area.
A=xy and 2x+y=63; this second equation is for the fencing length.
You want the maximum A and this you might guess if you look at the two equations, would be a quadratic function.  You can see I am using y as the length of the enclosure which is against the house.


{{{y=-2x+63}}} is from the fencing length equation;
{{{A=x*(-2x+63)}}}
{{{A=-2x^2+63x}}}
!.................That function is a parabola, opening downward, so it has a maximum point.  You can find the roots (the values of x for which A is zero) and the x value in the exact middle of those roots will occur at the maximum point of A.


Here we go:   {{{A=0=-2x^2+63x}}}
{{{0=x(-2x+63)}}}
One possibility for x is {{{highlight_green(x=0)}}}.
Other possibility for x is {{{-2x+63=0}}}
{{{2x=63}}}
{{{x=63/2}}}
{{{highlight_green(x=31&1/2=63/2)}}}
'
The x value in the middle:  {{{(63/2)/2=highlight(63/4=15&3/4)}}}


Going back to the formula for y derived from the fence length above, you should find that y and x are the same value; so the maximum area is a square.