AHSME 1973 Problem 6

If 554 is the base $b$ representation of the square of the number whose base $b$ representation is 24, then $b$, when written in base 10, equals $\textbf{(A)}\ 6 \textbf{(B)}\ 8 \textbf{(C)}\ 12 \textbf{(D)}\ 14 \textbf{(E)}\ 16$