Answer
If a cylindrical soft drink has a volume of $12$ fluid ounces, approximately $22\text{ cubic}\text{ }\text{inches}$, then the surface area of the can, $A$ in square inches as a function of its radius $r$ is given as:
$A\left( r \right)=2\pi {{r}^{2}}+\frac{44}{r}$.
Work Step by Step
We have a cylindrical can.
The volume of the can is:
$\text{V=}\pi {{r}^{2}}h$
And the volume is $22\text{~cubic}\text{ }\text{inches}$
Therefore,
$\begin{align}
& \text{22=}\pi {{r}^{2}}h \\
& h=\frac{22}{\pi {{r}^{2}}}
\end{align}$
The area of the can: $A=2\pi {{r}^{2}}+2\pi rh$
Now put the value of $h$ in the above equation,
$\begin{align}
& A=2\pi {{r}^{2}}+2\pi rh \\
& =2\pi {{r}^{2}}+2\pi r\left( \frac{22}{\pi {{r}^{2}}} \right) \\
& =2\pi {{r}^{2}}+\frac{44}{r}
\end{align}$
Here, $A$ is the function of $r$ (can radius),
Hence $A$ is expressed as,
$A\left( r \right)=2\pi {{r}^{2}}+\frac{44}{r}$
