jmc

Let the original radius be $r$. The volume of the cylinder with the increased radius is $\pi \cdot (r+6{)}^2\cdot 2$. The volume of the cylinder with the increased height is $\pi \cdot r^2\cdot 8$. Since we are told these two volumes are the same, we have the equation $\pi \cdot (r+6{)}^2\cdot 2=\pi \cdot r^2\cdot 8$. Simplifying, we have $(r+6{)}^2=4r^2\Rightarrow r^2+12r+36=4r^2\Rightarrow r^2-4r-12=0$. Factoring, we have $(r-6)(r+2)=0$, so $r=\boxed{6}$ (it cannot be negative because it is a physical length).