8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{659a0479-c8c6-418b-b8a9-67ad68474023-18_367_709_280_676}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
A bowl is modelled as a hemispherical shell as shown in Figure 3.
Initially the bowl is empty and water begins to flow into the bowl.
When the depth of the water is \(h \mathrm {~cm}\), the volume of water, \(V \mathrm {~cm} ^ { 3 }\), according to the model is given by
$$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 75 - h ) , \quad 0 \leqslant h \leqslant 24$$
The flow of water into the bowl is at a constant rate of \(160 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) for \(0 \leqslant h \leqslant 12\)
- Find the rate of change of the depth of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 10\)
Given that the flow of water into the bowl is increased to a constant rate of \(300 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) for \(12 < h \leqslant 24\)
- find the rate of change of the depth of the water, in \(\mathrm { cms } ^ { - 1 }\), when \(h = 20\)