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:
$\textbf{(A)}\ \text{no real value of x} \qquad \\ \textbf{(B)}\ \text{one integral value of x} \qquad \\ \textbf{(C)}\ \text{one rational, but not integral, value of x} \qquad \\ \textbf{(D)}\ \text{one irrational value of x}\qquad \\ \textbf{(E)}\ \text{two real values of x}$
Since increasing the height by $x$ inches should result in the same volume as increasing the radius by $x$ inches, write an equation with the two cylinders (one with height increased, one with radius increased).
\[\pi (8+x)^2 \cdot 3 = \pi \cdot 8^2 \cdot (3+x)\]
\[3(x^2+16x+64) = 64(x+3)\]
\[3x^2+48x+192=64x+192\]
\[3x^2-16x=0\]
\[x(3x-16)=0\]
By the Zero-Product Property, $x = 0$ or $\frac{16}{3}$.  However, since $x$ must be increased, discard $x = 0$ as a possible value.  Thus, the length should be increased by $\frac{16}{3}$ inches, so the answer is $\boxed{\textbf{(C)}}$.