6.
\section*{In this question you must show all stages of your working.}
\section*{Solutions relying entirely on calculator technology are not acceptable.}
- Express \(3 \cos x + \sin x\) in the form \(R \cos ( x - \alpha )\) where
- \(\quad R\) and \(\alpha\) are constants
- \(R > 0\)
- \(0 < \alpha < \frac { \pi } { 2 }\)
Give the exact value of \(R\) and the value of \(\alpha\) in radians to 3 decimal places.
The temperature, \(\theta ^ { \circ } \mathrm { C }\), inside a rabbit hole on a particular day is modelled by the equation
$$\theta = 6.5 + 3 \cos \left( \frac { \pi t } { 13 } - 4 \right) + \sin \left( \frac { \pi t } { 13 } - 4 \right) \quad 0 \leqslant t < 24$$
where \(t\) is the number of hours after midnight.
Using the equation of the model and your answer to part (a) - deduce the minimum value of \(\theta\) during this day,
- find the time of day when this minimum value occurs, giving your answer to the nearest minute.
- Find the rate of temperature increase in the rabbit hole at midday.
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