smc

Because $P$ varies jointly as $A$ and $V^2$, that means that there is a number $k$ such that $P=kA{V}^2$. You are given that $P=1$ when $A=1$ and $V=16$. That means that $1=k(1)(16^2)\to k=\frac{1}{256}$. Then, substituting into the original equation with $P=36$ and $A=9$ (because a square yard is $9$ times a square foot), you get $4=\frac{1}{256}(V^2)$. Solving for $V$, we get $V^2=1024$, so $V=32$. Hence, the answer is $\boxed{32\text{ mph}}$.