Numeracy Higher June 2019 Unit 2 Q9
9. Alpha, Beta and Gamma are three boats.
They receive a weather warning and need to go to the port of Aberwyn.
The following diagram shows the positions of the three boats when the weather warning is received.
Diagram not drawn to scale
From the formula list: In any triangle \(ABC\): Sine rule \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\); Cosine rule \(a^2 = b^2 + c^2 - 2bc\cos A\)
The captains of Alpha and Beta need to know their distances from Aberwyn in order to find how long it will take them to get to the port.
Calculate the distance of each of the boats Alpha and Beta from Aberwyn. [7]
| Scheme | Marks | Comments |
|---|---|---|
| Use of cosine rule followed by sine rule | S1 | |
| Distance of Alpha from Aberwyn (distance =) \(\sqrt{5.5^2 + 2.4^2 - 2 \times 5.5 \times 2.4 \times \cos 76(^\circ)}\) or (dist =) √29.623… | M2 | M1 for (distance2 =) 5.52 + 2.42 – 2×5.5×2.4×cos76(º) or (dist2 =) 29.623… |
| (distance =) 5.4(427…) (km) | A1 | CAO |
| Distance of Beta from Aberwyn (distance =) \(\dfrac{5.4(427\ldots) \times \sin 32(^\circ)}{\sin 118(^\circ)}\) | M2 | FT ‘their derived 5.4(427…)’ M1 for \(\dfrac{\text{distance}}{\sin 32(^\circ)} = \dfrac{5.4(427\ldots)}{\sin 118(^\circ)}\) |
| = 3.2(4…) to 3.3 (km) | A1 | FT from M2 for their sine rule only |