1951 AHSME 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\qquad\textbf{(B)}\ 5\frac{1}{3}\qquad\textbf{(C)}\ \text{any number}\qquad\textbf{(D)}\ \text{non-existent}\qquad\textbf{(E)}\ \text{none of these}

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