M4 June 2014 (R) Q2
2. A ship \(A\) is travelling at a constant speed of 30 km h\(^{-1}\) on a bearing of 050\(^\circ\). Another ship \(B\) is travelling at a constant speed of \(v\) km h\(^{-1}\) and sets a course to intercept \(A\). At 1400 hours \(B\) is 20 km from \(A\) and the bearing of \(A\) from \(B\) is 290\(^\circ\).
(a) Find the least possible value of \(v\). (3)
Given that \(v = 32\),
(b) find the time at which \(B\) intercepts \(A\). (8)
| Scheme | Marks |
|---|---|
| Vector triangle with perpendicular | M1 |
| Minimum speed \(= 30\sin 60^\circ\) | M1 |
| \(= 15\sqrt{3}\) oe \(\quad (26.0)\) | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\dfrac{\sin\theta}{30} = \dfrac{\sin 60^\circ}{32}\) | M1 A1 |
| \(\Rightarrow \theta = 54.281\ldots^\circ\) | A1 |
| \(v^2 = 32^2 + 30^2 - 2.32.30\cos(120^\circ - \theta)\) | M1 A1 |
| \(= 33.68\ldots \qquad \left(15 + \sqrt{349}\right)\) | A1 |
| \(t = \dfrac{20}{33.68\ldots} = 0.59379\ldots\text{h} = 35.62\ldots\text{min}\) | M1 |
| Time is 2.36 pm (14.36) | A1 |
| (8) | |
| (11 marks) |