jmc

Since $a$ varies inversely with $b^2$, $(a)(b^2)=k$ for some constant $k$. If $a=9$ when $b=2$, then $k=(9)(2^2)=(9)(4)=36$. So if $b=3$, $$\begin{array}{rl}(a)(3^2)& =36\\ 9a& =36\\ \Rightarrow a& =\boxed{4}\end{array}$$