AMC 12 2005a 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 $r$?

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