jmc

Since $\cos R=\frac{4}{9}$ and $\cos R=\frac{QR}{RS}=\frac{QR}{9}$, we have $\frac{QR}{9}=\frac{4}{9}$, so $QR=4$. Then, by the Pythagorean Theorem, $QS=\sqrt{R{S}^2-Q{R}^2}=\sqrt{81-16}=\boxed{\sqrt{65}}$.