AHSME 1983 Problem 27

A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.) $\textbf{(A)}\ \frac{5}{2} \textbf{(B)}\ 9 - 4\sqrt{5} \textbf{(C)}\ 8\sqrt{10} - 23 \textbf{(D)}\ 6-\sqrt{15} \textbf{(E)}\ 10\sqrt{5}-20$