Question 142358
Let  one side = {{{c}}}
Let the other side= {{{d}}}
Let area = {{{A}}}
Let perimeter= {{{P}}}
{{{A = cd}}}
{{{P = 2c + 2d}}}
{{{22 = 2c + 2d}}}
{{{11 = c + d}}}
{{{d = 11 - c}}}
Now substitute into area formula
{{{A = c*(11 - c)}}}
{{{A = -c^2 + 11c}}}
This is a parabola that has a peak. I know because
of the (-) sign in front of {{{c^2}}}
The peak is exactly between the x-intercepts, or at
{{{c = (-b)/(2a)}}} where {{{a = -1}}} and {{{b = 11}}}
because the equation is in the form {{{A = ac^2 + bc}}}
{{{c = (-11)/(-2)}}}
{{{c = 11/2}}}
{{{P = 2(11/2) + 2d}}}
{{{22 = 11 + 2d}}}
{{{d = 11/2}}}
So, the max area is 
{{{A[max] = (11/2)(11/2)}}}
{{{A[max] = 121/4}}}
{{{A[max] = 30.25}}}