C3 January 2009 Q8
8.
(a) Express \(3\cos\theta + 4\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90^\circ\). (4)
(b) Hence find the maximum value of \(3\cos\theta + 4\sin\theta\) and the smallest positive value of \(\theta\) for which this maximum occurs. (3)
The temperature, \(\mathrm{f}(t)\), of a warehouse is modelled using the equation\[\mathrm{f}(t) = 10 + 3\cos(15t)^\circ + 4\sin(15t)^\circ,\]where \(t\) is the time in hours from midday and \(0 \leqslant t < 24\).
(c) Calculate the minimum temperature of the warehouse as given by this model. (2)
(d) Find the value of \(t\) when this minimum temperature occurs. (3)
| Scheme | Marks |
|---|---|
| \(R^2 = 3^2 + 4^2\) | M1 |
| \(R = 5\) | A1 |
| \(\tan\alpha = \dfrac{4}{3}\) | M1 |
| \(\alpha = 53\ \ldots^\circ\) awrt \(53^\circ\) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| Maximum value is 5 ft their \(R\) | B1 ft |
| At the maximum, \(\cos(\theta - \alpha) = 1\) or \(\theta - \alpha = 0\) | M1 |
| \(\theta = \alpha = 53\ \ldots^\circ\) ft their \(\alpha\) | A1 ft |
| (3) |
| Scheme | Marks |
|---|---|
| \(\mathrm{f}(t) = 10 + 5\cos(15t - \alpha)^\circ\) Minimum occurs when \(\cos(15t - \alpha)^\circ = -1\) | M1 |
| The minimum temperature is \((10 - 5)^\circ = 5^\circ\) | A1 ft |
| (2) |
| Scheme | Marks |
|---|---|
| \(15t - \alpha = 180\) | M1 |
| \(t = 15.5\) awrt 15.5 | M1 A1 |
| (3) | |
| (12 marks) |