jmc

Since $a^2$ and $\sqrt{b}$ are inversely proportional, $a^2\sqrt{b}=k$ for some constant k. Thus $k=2^2\sqrt{81}=36$. Squaring both sides gives $a^4\cdot b=1296$, so if $ab=48$ then dividing those two equations gives $a^3=\frac{1296}{48}=27$, so $a=3$ and $b=\frac{48}{3}=\boxed{16}$.