15.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-23_609_493_223_762}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
Figure 5 shows a design for a water barrel.
It is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and radius \(r \mathrm {~cm}\).
The barrel has a base but has no lid, is open at the top and is made of material of negligible thickness.
The barrel is designed to hold \(60000 \mathrm {~cm} ^ { 3 }\) of water when full.
- Show that the total external surface area, \(S \mathrm {~cm} ^ { 2 }\), of the barrel is given by the formula
$$S = \pi r ^ { 2 } + \frac { 120000 } { r }$$
- Use calculus to find the minimum value of \(S\), giving your answer to 3 significant figures.
- Justify that the value of \(S\) you found in part (b) is a minimum.