2017 Fermat Problem 22

A cylinder has radius 12 and height 30. The top circular face of the cylinder is the base of a cone and the centre of the bottom circular base of the cylinder is the vertex of the cone. A sphere is placed inside so that it touches the cone, the base of the cylinder and the side of the cylinder as shown. Which of the following is closest to the radius of the sphere?

\textbf{(A)}\ 4.84\quad \textbf{(B)}\ 4.74\quad \textbf{(C)}\ 4.64\quad \textbf{(D)}\ 4.54\quad \textbf{(E)}\ 4.44

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.