#### Answer

$a=4$

#### Work Step by Step

Recall:
If $y$ varies directly as the $n^{\text{th}}$ power of $x$, then the direct variation's equation is $y=kx^n$ where $k$ is the constant of variation.
Since $a$ varies directly as the square of $b$, then the equation of the direct variation, with $k$ as the constant of variation, is:
$$a=kb^2$$
When $a=4$, $b=3$. Substitute these into the equation above to obtain:
\begin{align*}
a&=kb^2\\\\
4&=k(3^2)\\\\
4&=9k\\\\
\frac{4}{9}&=k
\end{align*}
Thus, the equation fo the direct variation is:
$$a=\frac{4}{9}b^2$$
To find the value of $a$ when $b=2$, substitute $2$ to $b$ in the equation above to obtain:
\begin{align*}
a&=\frac{4}{9}b^2\\\\
a&=\frac{4}{9} \cdot (3^2)\\\\
a&=\frac{4}{9} \cdot 9\\\\
a&=4
\end{align*}