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 $y$.



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