7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e116a86f-63e0-4e80-b49c-d9f3c819ce15-14_495_711_243_641}
\captionsetup{labelformat=empty}
\caption{Diagram not drawn to scale.}
\end{figure}
Figure 2
Figure 2 shows a cylindrical tank of height 1.5 m .
Initially the tank is full of water.
The water starts to leak from a small hole, at a point \(L\), in the side of the tank.
While the tank is leaking, the depth, \(H\) metres, of the water in the tank is modelled by the differential equation
$$\frac { \mathrm { d } H } { \mathrm {~d} t } = - 0.12 \mathrm { e } ^ { - 0.2 t }$$
where \(t\) hours is the time after the leak starts.
Using the model,
- show that
$$H = A \mathrm { e } ^ { - 0.2 t } + B$$
where \(A\) and \(B\) are constants to be found,
- find the time taken for the depth of the water to decrease to 1.2 m . Give your answer in hours and minutes, to the nearest minute.
In the long term, the water level in the tank falls to the same height as the hole.
- Find, according to the model, the height of the hole from the bottom of the tank.