Precalculus (6th Edition) Blitzer

The expression for the cross-sectional area of the rain gutter as a function of its depth x is $A\left( x \right)=-2{{x}^{2}}+20x$
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 $A\left( x \right)=-2{{x}^{2}}+20x$