7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fe01157a-7617-43d3-900c-8d043bcbe784-12_252_757_267_484}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a hemispherical bowl of radius 5 cm .
The bowl is filled with water but the water leaks from a hole at the base of the bowl. At time \(t\) minutes, the depth of water is \(h \mathrm {~cm}\) and the volume of water in the bowl is \(V \mathrm {~cm} ^ { 3 }\), where
$$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 15 - h ) .$$
In a model it is assumed that the rate at which the volume of water in the bowl decreases is proportional to \(V\).
- Show that
$$\frac { \mathrm { d } h } { \mathrm {~d} t } = - \frac { k h ( 15 - h ) } { 3 ( 10 - h ) } ,$$
where \(k\) is a positive constant.
- Express \(\frac { 3 ( 10 - h ) } { h ( 15 - h ) }\) in partial fractions.
Given that when \(t = 0 , h = 5\),
- show that
$$h ^ { 2 } ( 15 - h ) = 250 \mathrm { e } ^ { - k t }$$
Given also that when \(t = 2 , h = 4\),
- find the value of \(k\) to 3 significant figures.
7. continued
7. continued