Single Variable Calculus, Chapter 1, 1.1, Section 1.1, Problem 54

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




Express the surface area of a cube as a function of its volume.




The volume of the cube is equal to the product of its length, width and height where all the sides are equal.





$\begin{equation} \begin{aligned} \text{Volume } &= s^3 && ; \text{where } s \text{ is the edge of the cube}\\ s &= \sqrt[3]{\text{Volume}}&& (\text{Solving for } s) \end{aligned} \end{equation}$





The surface area of the cube is equal to the sum of the areas of each faces of the cube and is equal to...




$\begin{equation} \begin{aligned} \text{Surface Area} &= 6s^2 \end{aligned} \end{equation}$




Substituting the value of $s$ to the surface area, we get...




$\begin{equation} \begin{aligned} \text{Surface Area} = 6(\sqrt[3]{\text{Volume}})^2\\ \fbox{$\text{Surface Area} = 6(\text{Volume})^{\frac{2}{3}}$} \end{aligned} \end{equation}$