13.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa4afaf4-fe5d-4f3a-b3de-9600d5502a49-40_501_401_242_831}
\captionsetup{labelformat=empty}
\caption{Figure 9}
\end{figure}
[A sphere of radius \(r\) has volume \(\frac { 4 } { 3 } \pi r ^ { 3 }\) and surface area \(4 \pi r ^ { 2 }\) ]
A manufacturer produces a storage tank.
The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9.
The walls of the tank are assumed to have negligible thickness.
The cylinder has radius \(r\) metres and height \(h\) metres and the hemisphere has radius \(r\) metres.
The volume of the tank is \(6 \mathrm {~m} ^ { 3 }\).
- Show that, according to the model, the surface area of the tank, in \(\mathrm { m } ^ { 2 }\), is given by
$$\frac { 12 } { r } + \frac { 5 } { 3 } \pi r ^ { 2 }$$
The manufacturer needs to minimise the surface area of the tank.
- Use calculus to find the radius of the tank for which the surface area is a minimum.
(4) - Calculate the minimum surface area of the tank, giving your answer to the nearest integer.