8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e29d66c-c3c6-4e4b-acfb-c73c60604d93-11_453_1225_255_369}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Kate crosses a road, of constant width 7 m , in order to take a photograph of a marathon runner, John, approaching at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Kate is 24 m ahead of John when she starts to cross the road from the fixed point \(A\). John passes her as she reaches the other side of the road at a variable point \(B\), as shown in Figure 2.
Kate's speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and she moves in a straight line, which makes an angle \(\theta\), \(0 < \theta < 150 ^ { \circ }\), with the edge of the road, as shown in Figure 2.
You may assume that \(V\) is given by the formula
$$V = \frac { 21 } { 24 \sin \theta + 7 \cos \theta } , \quad 0 < \theta < 150 ^ { \circ }$$
- Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants and where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) to 2 decimal places.
Given that \(\theta\) varies,
- find the minimum value of \(V\).
Given that Kate's speed has the value found in part (b),
- find the distance \(A B\).
Given instead that Kate's speed is \(1.68 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
- find the two possible values of the angle \(\theta\), given that \(0 < \theta < 150 ^ { \circ }\).