1960 AHSME Problem 23

The radius R of a cylindrical box is 8 inches, the height H is 3 inches. The volume V = \pi R^2H is to be increased by the same fixed positive amount when R is increased by x inches as when H is increased by x inches. This condition is satisfied by:

\begin{array}{l} \textbf{(A)}\ \text{no real value of x} \qquad \\ \textbf{(B)}\ \text{one integral value of x} \qquad \\ \textbf{(C)}\ \text{one rational, but not integral, value of x} \qquad \\ \textbf{(D)}\ \text{one irrational value of x}\qquad \\ \textbf{(E)}\ \text{two real values of x} \end{array}

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