Answer
The perimeter of a rectangle is 80 feet. If the length of the rectangle is represented by x, its width can be expressed as $40-x$. The area of the rectangle, $A\left( x \right)$, expressed in the form $A\left( x \right)=a{{x}^{2}}+bx+c$, is $A\left( x \right)=-{{x}^{2}}+40x$.
Work Step by Step
The perimeter of a rectangle is 80 feet. It is assumed that the length of the rectangle is x and the width of the rectangle is y.
The perimeter of the rectangle is:
$\text{Perimeter}=2\left( \text{length}+\text{width} \right)$
Therefore:
$\begin{align}
& \text{perimeter}=2x+2y \\
& 80=2\left( x+y \right) \\
& 40=x+y \\
& 40-x=y
\end{align}$
So, the width of rectangle is $y=40-x$.
The area of rectangle,
$\begin{align}
& \text{Area}=\text{Length}\times \text{Width} \\
& =x\cdot y
\end{align}$
The area of the rectangle is calculated as below:
$\begin{align}
& A\left( x \right)=xy \\
& =x\left( 40-x \right) \\
& =-{{x}^{2}}+40x
\end{align}$
So, the product of $A\left( x \right)$ is expressed in the form $A\left( x \right)=a{{x}^{2}}+bx+c$ is $-{{x}^{2}}+40x$.
