8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1ef99f0-4ad4-49d8-bee7-d5bb9cc84660-11_305_446_223_749}
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\caption{Figure 3}
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A manufacturer produces pain relieving tablets. Each tablet is in the shape of a solid circular cylinder with base radius \(x \mathrm {~mm}\) and height \(h \mathrm {~mm}\), as shown in Figure 3.
Given that the volume of each tablet has to be \(60 \mathrm {~mm} ^ { 3 }\),
- express \(h\) in terms of \(x\),
- show that the surface area, \(A \mathrm {~mm} ^ { 2 }\), of a tablet is given by \(A = 2 \pi x ^ { 2 } + \frac { 120 } { x }\)
The manufacturer needs to minimise the surface area \(A \mathrm {~mm} ^ { 2 }\), of a tablet.
- Use calculus to find the value of \(x\) for which \(A\) is a minimum.
- Calculate the minimum value of \(A\), giving your answer to the nearest integer.
- Show that this value of \(A\) is a minimum.