2009 Pascal Problem 24

Eight identical spheres, each of diameter 20, fit tightly into a cube of side length 40 so that each sphere just touches three of the faces of the cube. The radius of the largest sphere that will fit in the central space, just touching all eight spheres, is closest to

\textbf{(A)}\ 7.0\quad \textbf{(B)}\ 7.3\quad \textbf{(C)}\ 7.6\quad \textbf{(D)}\ 7.9\quad \textbf{(E)}\ 8.2

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 Pascal contests here.