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}
$