3 Fig. 8.1 shows an upright cylindrical barrel containing water. The water is leaking out of a hole in the side of the barrel.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{11d26af4-19d0-4310-a64e-9888285c9980-2_260_447_281_824}
\captionsetup{labelformat=empty}
\caption{Fig. 8.1}
\end{figure}
The height of the water surface above the hole \(t\) seconds after opening the hole is \(h\) metres, where
$$\frac { \mathrm { d } h } { \mathrm {~d} t } = - A \sqrt { h }$$
and where \(A\) is a positive constant. Initially the water surface is 1 metre above the hole.
- Verify that the solution to this differential equation is
$$h = \left( 1 - \frac { 1 } { 2 } A t \right) ^ { 2 } .$$
The water stops leaking when \(h = 0\). This occurs after 20 seconds.
- Find the value of \(A\), and the time when the height of the water surface above the hole is 0.5 m .
Fig. 8.2 shows a similar situation with a different barrel; \(h\) is in metres.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{11d26af4-19d0-4310-a64e-9888285c9980-2_235_455_1425_820}
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\caption{Fig. 8.2}
\end{figure}
For this barrel,
$$\frac { \mathrm { d } h } { \mathrm {~d} t } = - B \frac { \sqrt { h } } { ( 1 + h ) ^ { 2 } } ,$$
where \(B\) is a positive constant. When \(t = 0 , h = 1\). - Solve this differential equation, and hence show that
$$h ^ { \frac { 1 } { 2 } } \left( 30 + 20 h + 6 h ^ { 2 } \right) = 56 - 15 B t .$$
- Given that \(h = 0\) when \(t = 20\), find \(B\).
Find also the time when the height of the water surface above the hole is 0.5 m .