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Math SL Bearing Question

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A ship is sailing due west when the captain sees a lighthouse at a distance of 20 km on a bearing of 230°.

a. how far must the ship sail before the lighthouse is 16km away? 

b. how far must the ship sail beyond this point before the lighthouse is again at a distance of 16km from the ship?

c. what is the bearing of the lighthouse from the ship the second time the two are 16km apart?

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Let the lighthouse be at the origin O(0, 0). From the lighthouse perspective, the ship's initial position A(20 sin 50°, 20 cos 50°) at a bearing of (230-180 =) 50°. B and C are the locations that make the distance 16 km. See diagram.

O1cwOPU.jpg

a) b) Travelling westward only decreases the x coordinate. After travelling for d km, d > 0, the position is B(20 sin 50° - d, 20 cos 50°), which is 16 km away from the origin. So 16² = (20 sin 50° - d)² + (20 cos 50°)² = 400 sin² 50° - 40d sin 50° + d² + 400 cos² 50°
d² - 30.642d + 144 = 0
d = 5.796, 24.846

So ship needs to travel 5.80 km before 16 km away from lighthouse, and another (24.846 - 5.796 = )19.05 km before 16 km away from lighthouse again

c) θ = arccos ((20 cos 50°)/16) = 36.5°. The ship is -36.5° from the lighthouse, so the lighthouse is (180 - 36.5 =) 143.5° from the ship.

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