Single Variable Calculus, Chapter 1, 1.1, Section 1.1, Problem 51

A rectangle has perimeter of 20$m$. Express the area of the rectangle as a function of the length of one of its sides.

The perimeter of the rectangle is the sum of its length and width. As shown below:


$
\begin{equation}
\begin{aligned}
\text{Perimeter} &= 2x + 2y && ;\text{where } x \text{ and } y \text{ are the length and width respectively.}\\
2x + 2y &= 20 && (\text{Dividing both sides of the equation by 2})\\
x+y &= 10 && (\text{Solving for } y)\\
y &= 10 - x\\
\end{aligned}
\end{equation}
$


While the area of the rectangle is the product of its length and width.



$
\begin{equation}
\begin{aligned}
\text{Area} &= xy && (\text{Substituting the value of } y \text{ from the Perimeter})\\
\end{aligned}
\end{equation}
$




$
\begin{equation}
\begin{aligned}
\boxed{
\begin{array}{lll}
&\text{Area} &=& x(10-x) &&\\
&\text{domain:} && 0 < x < 10 &&
\end{array}}

\end{aligned}
\end{equation}
$