2011 Fermat Problem 25

A pyramid has a square base with side length 20. A right circular cylinder has a diameter of 10 and a length of 10. The cylinder is lying on its side, completely inside the pyramid. The central axis of the cylinder lies parallel to and directly above a diagonal of the pyramid's base. The midpoint of the central axis lies directly above the centre of the square base of the pyramid.

The smallest possible height of the pyramid is closest to

\textbf{(A)}\ 15.3\quad \textbf{(B)}\ 22.1\quad \textbf{(C)}\ 21.9\quad \textbf{(D)}\ 21.7\quad \textbf{(E)}\ 15.5

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Fermat contests here.