12.
$$\mathrm { f } ( x ) = 10 \mathrm { e } ^ { - 0.25 x } \sin x , \quad x \geqslant 0$$
- Show that the \(x\) coordinates of the turning points of the curve with equation \(y = \mathrm { f } ( x )\) satisfy the equation \(\tan x = 4\)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-34_687_1029_495_518}
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\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\). - Sketch the graph of \(H\) against \(t\) where
$$\mathrm { H } ( t ) = \left| 10 \mathrm { e } ^ { - 0.25 t } \sin t \right| \quad t \geqslant 0$$
showing the long-term behaviour of this curve.
The function \(\mathrm { H } ( t )\) is used to model the height, in metres, of a ball above the ground \(t\) seconds after it has been kicked.
Using this model, find
- the maximum height of the ball above the ground between the first and second bounce.
- Explain why this model should not be used to predict the time of each bounce.