smc

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 $\boxed{\text{one rational, but not integral, value of x}}$.