9. Two circles are concentric. The area of the ring between them is $A$. In terms of $A$, find the length of the longest chord contained entirely within the ring.
Solution: Let the radii of the circles be $r$ and $R>r$, so $A=\pi\left(R^{2}-r^{2}\right)$. By the Pythagorean theorem, the length of the chord is $2 \sqrt{R^{2}-r^{2}}=2 \sqrt{\frac{A}{\pi}}$.