15.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{48f9a252-61a2-491d-94d0-8470aee96942-22_750_1100_276_541}
\captionsetup{labelformat=empty}
\caption{Figure 7}
\end{figure}
Figure 7 shows an open tank for storing water, \(A B C D E F\). The sides \(A C D F\) and \(A B E F\) are rectangles. The faces \(A B C\) and \(F E D\) are sectors of a circle with radius \(A B\) and \(F E\) respectively.
- \(A B = F E = r \mathrm {~cm}\)
- \(A F = B E = C D = l \mathrm {~cm}\)
- angle \(B A C =\) angle \(E F D = 0.9\) radians
Given that the volume of the tank is \(360 \mathrm {~cm} ^ { 3 }\)
a. show that the surface area of the tank, \(S \mathrm {~cm} ^ { 2 }\), is given by
$$S = 0.9 r ^ { 2 } + \frac { 1600 } { r }$$
(4)
Given that \(r\) can vary
b. use calculus to find the value of \(r\) for which \(S\) is stationary.
c. Find, to 3 significant figures the minimum value of \(S\).