Volume of revolution leading to a cubic in k

The curve with equation $y = \sqrt{k - \frac{1}{x^2}}$, for $1 \le x \le \frac{k}{2}$ where $k > 2$, is rotated about the $x$-axis to form a solid of revolution that has volume $\frac{7\pi}{2}$ units$^3$.

Show that $k$ satisfies the equation $k^3 - 2k^2 - 9k + 4 = 0$.