Single Variable Calculus, Chapter 1, 1.1, Section 1.1, Problem 52

The problem below describes a function. Find its formula and domain.


A rectangle has area 16 $m^2$. Express the perimeter of the rectangle as a function of the length of one of its size.

The area of rectangle is the product of its sides.



$
\begin{equation}
\begin{aligned}
\text{Area }&= xy &&;\text{where } x \text{ and } y \text{ are the sides of the rectangle}\\
xy &= 16 && (\text{Solving for } y)\\
y &= \frac{16}{x}
\end{aligned}
\end{equation}
$



The perimeter of the rectangle is equal to the sum of its sides and is equal to...



$
\begin{equation}
\begin{aligned}
\text { Perimeter } &= 2x + 2y && (\text{Substituting the value of } y)\\
\text { Perimeter } &= 2x + 2 \left( \frac{16}{x} \right)\\
\text { Perimeter } &= 2x + \frac{32}{x} && (\text{Simplifying the equation})\\

\end{aligned}
\end{equation}
$




$

\fbox{ $\begin{array}{cc}
\text{ Perimeter} = & \displaystyle \frac{2x^2 + 32}{x} \\
\text{domain} : & ( 0 , \infty)
\end{array} $}

$