Answer
Depth $3$ inches, wide $6$ inches, and cross-section area is $18$ sq. inches.
Work Step by Step
We need to maximize the cross-sectional area of the rain gutter
$\begin{align}
& A\left( x \right)=x\left( 12-2x \right) \\
& =12x-2{{x}^{2}}
\end{align}$
Since $a=-2$ is negative, then the function opens downward and has a maximum at
$\begin{align}
& x=-\frac{b}{2a} \\
& =-\frac{12}{2\left( -2 \right)} \\
& =3
\end{align}$
Thus, cross-sectional is
$\begin{align}
& A\left( 3 \right)=12\left( 3 \right)-2{{\left( 3 \right)}^{2}} \\
& =36-18 \\
& =18
\end{align}$
Hence, the cross-sectional area is $18$ sq. inches.
