AHSME 1951 Problem 23

The radius of a cylindrical box is $8$ inches and the height is $3$ inches. The number of inches that may be added to either the radius or the height to give the same nonzero increase in volume is: $\textbf{(A)}\ 1 \textbf{(B)}\ 5\frac {1}{3} \textbf{(C)}\ \text{any number} \textbf{(D)}\ \text{non-existent} \textbf{(E)}\ \text{none of these}$