Let a rectangle have length and width .

Then by the first condition, we have .

Let . It follows from above that .

Substituting this into the area equation, we have .

Since we want to maximize area, it follows that . Now, we find the critical point:

(To verify its a max, we differentiate again to get )

Since , it follows that .

Therefore, the dimensions of the rectangle that gives us the maximum area actually form a square.

Does this make sense?