8. (a) Express \(9 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 4 decimal places.
(b) (i) State the maximum value of \(9 \cos \theta - 2 \sin \theta\) (ii) Find the value of \(\theta\), for \(0 < \theta < 2 \pi\), at which this maximum occurs. Ruth models the height \(H\) above the ground of a passenger on a Ferris wheel by the equation $$H = 10 - 9 \cos \left( \frac { \pi t } { 5 } \right) + 2 \sin \left( \frac { \pi t } { 5 } \right)$$ where \(H\) is measured in metres and \(t\) is the time in minutes after the wheel starts turning. \includegraphics[max width=\textwidth, alt={}, center]{0f6fd881-4d4b-4f80-96cc-6da41cc33c60-14_572_458_719_1158}
(c) Calculate the maximum value of \(H\) predicted by this model, and the value of \(t\), when this maximum first occurs. Give your answers to 2 decimal places.
(d) Determine the time for the Ferris wheel to complete two revolutions.

(a) $9\cos\theta - 2\sin\theta = R\cos(\theta + \alpha)$ | B1
$R = \sqrt{9^2 + 2^2} = \sqrt{85}$ | M1, A1
$\alpha = \arctan\left(\frac{2}{9}\right) = 0.21866... \approx 0.2187$ |

(b) (i) $\sqrt{85}$ | B1

(ii) $\theta + \alpha = 2\pi \Rightarrow \theta \approx 6.062$ (d.p.) | M1, A1

(c) Seeing (or implied by their working) $H = 10 - R\cos\left(\frac{\pi t}{5} + \alpha\right)$ for their $R$ and $\alpha$ | M1
$H_{\max} = 10 + \text{their } R = 10 + \sqrt{85} \approx 19.22$ m | A1
Maximum occurs when $\cos\left(\frac{\pi t}{5} + \alpha\right) = -1$ or $\frac{\pi t}{5} + \alpha = \pi$ | M1
$t \approx 4.65$ | A1

(d) Setting and solving $\frac{\pi t}{5} = 2\pi$ (for 1 cycle) or $\frac{\pi t}{5} = 4\pi$ (for 2 cycles) | M1
Two revolutions = 20 minutes | A1

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8. (a) Express $9 \cos \theta - 2 \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$.

Give the exact value of $R$ and give the value of $\alpha$ to 4 decimal places.\\
(b) (i) State the maximum value of $9 \cos \theta - 2 \sin \theta$\\
(ii) Find the value of $\theta$, for $0 < \theta < 2 \pi$, at which this maximum occurs.

Ruth models the height $H$ above the ground of a passenger on a Ferris wheel by the equation

$$H = 10 - 9 \cos \left( \frac { \pi t } { 5 } \right) + 2 \sin \left( \frac { \pi t } { 5 } \right)$$

where $H$ is measured in metres and $t$ is the time in minutes after the wheel starts turning.\\
\includegraphics[max width=\textwidth, alt={}, center]{0f6fd881-4d4b-4f80-96cc-6da41cc33c60-14_572_458_719_1158}\\
(c) Calculate the maximum value of $H$ predicted by this model, and the value of $t$, when this maximum first occurs. Give your answers to 2 decimal places.\\
(d) Determine the time for the Ferris wheel to complete two revolutions.

\hfill \mbox{\textit{Edexcel C3 2013 Q8 [12]}}