Answer
The expression for the cross-sectional area of the rain gutter as a function of its depth x is \[A\left( x \right)=-2{{x}^{2}}+20x\]
Work Step by Step
The gutter is made by folding the $x\text{ inches}$ of the sheet from both ends and thus the width of the gutter will be $20-x-x$ or $20-2x$.
The cross-sectional area of the gutter will be the product of the width of the gutter and depth of the gutter.
$\text{Cross section area}=\left[ \text{width of gutter} \right]\left[ \text{depth of gutter} \right]$
Substitute $20-2x$ for the width of the gutter, x for the depth of the gutter and $A\left( x \right)$ for the cross-sectional area.
$\begin{align}
& A\left( x \right)=\left( 20-2x \right)x \\
& =20x-2{{x}^{2}} \\
& =-2{{x}^{2}}+20x
\end{align}$
Hence, the expression for the cross-sectional area of the rain gutter as a function of its depth x is $A\left( x \right)=-2{{x}^{2}}+20x$
