7.
\begin{figure}[h]
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\caption{Figure 1}
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Figure 1 shows the curve \(C\), with equation \(y = 6 \cos x + 2.5 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\)
- Express \(6 \cos x + 2.5 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\) to 3 decimal places.
- Find the coordinates of the points on the graph where the curve \(C\) crosses the coordinate axes.
A student records the number of hours of daylight each Sunday throughout the year. She starts on the last Sunday in May with a recording of 18 hours, and continues until her final recording 52 weeks later.
She models her results with the continuous function given by
$$H = 12 + 6 \cos \left( \frac { 2 \pi t } { 52 } \right) + 2.5 \sin \left( \frac { 2 \pi t } { 52 } \right) , \quad 0 \leqslant t \leqslant 52$$
where \(H\) is the number of hours of daylight and \(t\) is the number of weeks since her first recording.
Use this function to find
- the maximum and minimum values of \(H\) predicted by the model,
- the values for \(t\) when \(H = 16\), giving your answers to the nearest whole number.
[0pt]
[You must show your working. Answers based entirely on graphical or numerical methods are not acceptable.]
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