- \hspace{0pt} [In this question you may assume the formula for the area of a circle and the following formulae:
a sphere of radius \(r\) has volume \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\) and surface area \(S = 4 \pi r ^ { 2 }\)
a cylinder of radius \(r\) and height \(h\) has volume \(V = \pi r ^ { 2 } h\) and curved surface area \(S = 2 \pi r h ]\)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ea81408b-e292-4529-b1e2-e3246503a3ac-25_414_478_566_726}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
Figure 5 shows the model for a building. The model is made up of three parts. The roof is modelled by the curved surface of a hemisphere of radius \(R \mathrm {~cm}\). The walls are modelled by the curved surface of a circular cylinder of radius \(R \mathrm {~cm}\) and height \(H \mathrm {~cm}\). The floor is modelled by a circular disc of radius \(R \mathrm {~cm}\). The model is made of material of negligible thickness, and the walls are perpendicular to the base.
It is given that the volume of the model is \(800 \pi \mathrm {~cm} ^ { 3 }\) and that \(0 < R < 10.6\)
- Show that
$$H = \frac { 800 } { R ^ { 2 } } - \frac { 2 } { 3 } R$$
- Show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of the model is given by
$$A = \frac { 5 \pi R ^ { 2 } } { 3 } + \frac { 1600 \pi } { R }$$
- Use calculus to find the value of \(R\), to 3 significant figures, for which \(A\) is a minimum.
- Prove that this value of \(R\) gives a minimum value for \(A\).
- Find, to 3 significant figures, the value of \(H\) which corresponds to this value for \(R\).