2005 AMC 12A Problem 22

A rectangular box P is inscribed in a sphere of radius r. The surface area of P is 384, and the sum of the lengths of its 12 edges is 112. What is r?

\textbf{(A) } 8 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 14 \qquad \textbf{(E) } 16

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.