1. Suppose, a rectangle circumscribes a quadrilateral having length of diagonals p and q, and area A.

What is the maximum area of rectangle that circumscribes the given quadrilateral?

How to answer this question using geometry or calculus or by using both techniques.

2.

3. Any other information about the "given" quadrilateral other than diagonal lengths, p and q, and area A?

4. If I can see correctly, no more information is needed. Just remember three details:

- Area of a quadrilateral is $\displaystyle A = \tfrac{1}{2}pq\sin \phi$, where $\displaystyle p$ and $\displaystyle q$ are the diagonal lengths and $\displaystyle \phi$ the angle between the diagonals.

- If you write a variable $\displaystyle z$ as $\displaystyle z = a + z - a$, nothing changes, but it might help solving considerably.

- Zeros of the function derivative gives you the critical points inside a closed interval; end points you need to check by substituting.

Hope this helps!

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