### Instruction: Suppose that $a^2$ varies inversely with $b^3$. If $a=7$ when $b=3$, find the value of $a^2$ when $b=6$.### Response: So we know that $a^2$ varies inversely with $b^3$.
That means $a^2=\frac{k}{b^3}$ for some constant k.
And we know that $a=7$ when $b=3$.
So $49=\frac{k}{27}$.
So $k=49*27$.
That means $a^2=\frac{49*27}{b^3}$.
Now we can plug in $b=6$ to find $a^2$.
So $a^2=\frac{49*27}{216}$.