jmc

A sphere with radius $r$ has volume $\frac{4}{3}\pi r^3$, so the volume of a hemisphere with radius $r$ is $\frac{2}{3}\pi r^3$. Therefore if a hemisphere of radius $r$ has the same volume as a sphere of radius $R$, we get $\frac{4}{3}\pi R^3=\frac{2}{3}\pi r^3$. Simplifying gives $R^3=\frac{1}{2}{r}^3\Rightarrow R=\frac{1}{\sqrt[3]{2}}r$. We know that $r=3\sqrt[3]{2}$ and that $R$ is the quantity we want to solve for, so substituting in our value of $r$ gives $R=\frac{1}{\sqrt[3]{2}}\cdot 3\sqrt[3]{2}=\boxed{3}.$