Question 690215
There is a side parallel to the house and there 
are 2 equal sides perpendicular to the house
Call each of the sides perpendicular 
to the house {{{ s }}}
The side parallel to the house is
{{{ 144 - 2s }}} ft long
The area, {{{ A }}} is
(1) {{{ A = s*( 144 - 2s ) }}}
(2) {{{ A = -2s^2 + 144s }}}
This is a parabola with a maximum, because of
the minus sign in front of the {{{ x^2 }}} term
The maximum is midway between the 2 roots
To find the roots:
(1) {{{ 0 = s*( 144 - 2s ) }}}
{{{ s = 0 }}}
{{{ 144 - 2s = 0 }}}
{{{ 2s = 144 }}}
{{{ s = 72 }}}
{{{ ( 72 + 0 ) / 2 = 36 }}}
The maximum is at (s,A) = (36,A)
Now find {{{ A }}}
{{{ A = -2*36^2 + 144*36 }}}
{{{ A = -2592 + 5184 }}}
{{{ A = 2592 }}}
The maximum area is 2592 ft2
check answer:
Here's the plot:
{{{ graph( 400, 400, -20, 80, -200, 2800, -2x^2 + 144x ) }}}