Single Variable Calculus, Chapter 1, 1.1, Section 1.1, Problem 55

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



An open rectangular box with volume 2 $m^3$ has a square base. Express the surface area of the box as a function of the length of a side of the base.









The volume of the figure can be written as the product of its length, width and height





$
\begin{equation}
\begin{aligned}
\text{Volume} &= x(x)(y)\\
\text{Volume} &= x^2 y\\
x^2y &= 2 && (\text{Solving for } y)\\
y &= \frac{2}{x^2}
\end{aligned}
\end{equation}
$





The total surface area of the box is equal to the sum of the areas of each faces of the figure.



$
\begin{equation}
\begin{aligned}
\text{Surface Area} = 4xy + x^2
\end{aligned}
\end{equation}
$




Now, we can relate the surface area of the box as the function of the length $x$ by substituting the value of $y$.



$
\begin{equation}
\begin{aligned}
\text{Surface Area} = 4x \left( \frac{2}{x^2}\right) + x^2\\
\boxed{\begin{array}{lll}
\text{Surface Area} & = & \displaystyle\frac{8}{x}+ x^2 \\
\text{domain:} & & x>0
\end{array} }
\end{aligned}
\end{equation}
$