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.



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




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