15.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{129adfbb-98fa-4e88-b636-7b4d111f3349-42_444_739_244_662}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
A company makes toys for children.
Figure 5 shows the design for a solid toy that looks like a piece of cheese.
The toy is modelled so that
- face \(A B C\) is a sector of a circle with radius \(r \mathrm {~cm}\) and centre \(A\)
- angle \(B A C = 0.8\) radians
- faces \(A B C\) and \(D E F\) are congruent
- edges \(A D , C F\) and \(B E\) are perpendicular to faces \(A B C\) and \(D E F\)
- edges \(A D , C F\) and \(B E\) have length \(h \mathrm {~cm}\)
Given that the volume of the toy is \(240 \mathrm {~cm} ^ { 3 }\)
- show that the surface area of the toy, \(S \mathrm {~cm} ^ { 2 }\), is given by
$$S = 0.8 r ^ { 2 } + \frac { 1680 } { r }$$
making your method clear.
Using algebraic differentiation,
- find the value of \(r\) for which \(S\) has a stationary point.
- Prove, by further differentiation, that this value of \(r\) gives the minimum surface area of the toy.