10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae871952-f525-44e6-8bac-09308aa1964f-38_570_671_310_680}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Diagram not drawn to scale
Figure 3 shows a container in the shape of an inverted right circular cone which contains some water.
The cone has an internal radius of 3 m and a vertical height of 5 m as shown in Figure 3.
At time \(t\) seconds,the height of the water is \(h\) metres,the volume of the water is \(V \mathrm {~m} ^ { 3 }\) and water is leaking from a hole in the bottom of the container at a constant rate of \(0.02 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
[The volume of a cone of radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) .]
- Show that,while the water is leaking,
$$h ^ { 2 } \frac { \mathrm {~d} h } { \mathrm {~d} t } = - \frac { 1 } { \mathrm { k } \pi }$$
where \(k\) is a constant to be found.
Given that the container is initially full of water,
- express \(h\) in terms of \(t\) .
- Find the time taken for the container to empty,giving your answer to the nearest minute.