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 
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 16$
Write out the numbers using the definition of [base numbers](https://artofproblemsolving.com/wiki/index.php/Base_numbers).
\[554_b = 5b^2 + 5b + 4\]
\[24_b = 2b+4\]
Since $554_b = (24_b)^2$, we can write an equation.
\[5b^2 + 5b + 4 = (2b+4)^2\]
\[5b^2 + 5b + 4 = 4b^2 + 16b + 16\]
\[b^2 - 11b - 12 = 0\]
\[(b-12)(b+1) = 0\]
Since base numbers must be positive, $b$ in base 10 equals $\boxed{\textbf{(C)}\ 12}$.