14.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-40_513_919_294_548}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
Water flows at a constant rate into a large tank.
The tank is a cuboid, with all sides of negligible thickness.
The base of the tank measures 8 m by 3 m and the height of the tank is 5 m .
There is a tap at a point \(T\) at the bottom of the tank, as shown in Figure 5.
At time \(t\) minutes after the tap has been opened
- the depth of water in the tank is \(h\) metres
- water is flowing into the tank at a constant rate of \(0.48 \mathrm {~m} ^ { 3 }\) per minute
- water is modelled as leaving the tank through the tap at a rate of \(0.1 h \mathrm {~m} ^ { 3 }\) per minute
- Show that, according to the model,
$$1200 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 24 - 5 h$$
Given that when the tap was opened, the depth of water in the tank was 2 m ,
show that, according to the model,
$$h = A + B \mathrm { e } ^ { - k t }$$
where \(A , B\) and \(k\) are constants to be found.
Given that the tap remains open,determine, according to the model, whether the tank will ever become full, giving a reason for your answer.