AHSME 1960 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: $\textbf{(A)}\ \text{no real value of x} \\ \textbf{(B)}\ \text{one integral value of x} \\ \textbf{(C)}\ \text{one rational, but not integral, value of x} \\ \textbf{(D)}\ \text{one irrational value of x} \\ \textbf{(E)}\ \text{two real values of x}$