Let and . The area of triangle BCD is , and the area of ABDE is . Hence the area of the pentagon P is
However you know that the perimeter is 10, so . Substitute into the area equation to obtain
Simplify, and differentiate both sides with respect to x and find critical points.
Suppose that the vertical sides of the rectangle each have length and suppose that the other sides have length
The area of the pentagon is
and the perimeter is
Substituting this into the area equation above produces
Differentiating,
We locate the critical value:
Now you can find