7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-24_835_1160_255_529}
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\caption{Figure 3}
\end{figure}
Figure 3 shows a vertical cylindrical tank of height 200 cm containing water. Water is leaking from a hole \(P\) on the side of the tank.
At time \(t\) minutes after the leaking starts, the height of water in the tank is \(h \mathrm {~cm}\).
The height \(h \mathrm {~cm}\) of the water in the tank satisfies the differential equation
$$\frac { \mathrm { d } h } { \mathrm {~d} t } = k ( h - 9 ) ^ { \frac { 1 } { 2 } } , \quad 9 < h \leqslant 200$$
where \(k\) is a constant.
Given that, when \(h = 130\), the height of the water is falling at a rate of 1.1 cm per minute,
- find the value of \(k\).
Given that the tank was full of water when the leaking started,
- solve the differential equation with your value of \(k\), to find the value of \(t\) when \(h = 50\)