9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08756c4b-6619-42da-ac8a-2bf065c01de8-30_528_1031_242_452}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a cylindrical tank that contains some water.
The tank has an internal diameter of 8 m and an internal height of 4.2 m .
Water is flowing into the tank at a constant rate of \(( 0.6 \pi ) \mathrm { m } ^ { 3 }\) per minute.
There is a tap at point \(T\) at the bottom of the tank.
At time \(t\) minutes after the tap has been opened,
- the depth of the water is \(h\) metres
- the water is leaving the tank at a rate of \(( 0.15 \pi h ) \mathrm { m } ^ { 3 }\) per minute
- Show that
$$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 12 - 3 h } { 320 }$$
Given that the depth of the water in the tank is 0.5 m when the tap is opened,
find the time taken for the depth of water in the tank to reach 3.5 m .