M4 June 2009 Q3
3. At noon a motorboat \(P\) is 2 km north-west of another motorboat \(Q\). The motorboat \(P\) is moving due south at 20 m s\(^{-1}\). The motorboat \(Q\) is pursuing motorboat \(P\) at a speed of 12 m s\(^{-1}\) and sets a course in order to get as close to motorboat \(P\) as possible.
(a) Find the course set by \(Q\), giving your answer as a bearing to the nearest degree. (4)
(b) Find the shortest distance between \(P\) and \(Q\). (3)
(c) Find the distance travelled by \(Q\) from its position at noon to the point of closest approach. (5)
| Scheme | Marks |
|---|---|
 | M1 |
| \(\cos\alpha = \tfrac{12}{20}\) | M1 A1 |
| Bearing is \(180^\circ + \alpha = 233^\circ\) (nearest degree) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(PN = 2000\cos(135^\circ - \alpha) = 200\sqrt{2}\) m or decimal equivalent | M1 A1ft A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\sqrt{20^2 - 12^2}\) | B1 |
| Time to closest approach \(= \dfrac{QN}{\sqrt{20^2 - 12^2}}\) | M1 |
| \(= \dfrac{2000\sin(135^\circ - \alpha)}{16}\) | A1 |
| Distance moved by \(Q\) = their \(t \times 12\) | DM1 |
| \(= 1050\sqrt{2}\) m or decimal equivalent | A1 |
| (5) | |
| (12 marks) |