Two ships are sailing in the sea on either side of a lighthouse. The angles of depression to the two ships as observed from the top of the lighthouse are $60^\circ$ and $45^\circ$, respectively. If the distance between the ships is $100\left(\dfrac{\sqrt{3}+1}{\sqrt{3}}\right)$ m, then find the height of the lighthouse.
Generated by claude-sonnet-4-6 · 2026-06-15 10:34 · grounding rag
Model Answer
Let the height of the lighthouse = h m.
Let the lighthouse be AB, with ships at C and D on either side.
- Angle of depression to ship C = 60° → ∠ACB = 60° (alternate angles)
- Angle of depression to ship D = 45° → ∠ADB = 45°
In right △ABC (angle 60°):
$$\tan 60° = \frac{h}{BC} \Rightarrow \sqrt{3} = \frac{h}{BC} \Rightarrow BC = \frac{h}{\sqrt{3}}$$
In right △ABD (angle 45°):
$$\tan 45° = \frac{h}{BD} \Rightarrow 1 = \frac{h}{BD} \Rightarrow BD = h$$
Total distance between ships:
$$BC + BD = \frac{h}{\sqrt{3}} + h = h\left(\frac{1}{\sqrt{3}} + 1\right) = h \cdot \frac{\sqrt{3}+1}{\sqrt{3}}$$
Given: $BC + BD = 100\left(\dfrac{\sqrt{3}+1}{\sqrt{3}}\right)$
$$h \cdot \frac{\sqrt{3}+1}{\sqrt{3}} = 100 \cdot \frac{\sqrt{3}+1}{\sqrt{3}}$$
$$\boxed{h = 100 \text{ m}}$$
The height of the lighthouse is 100 m.
Source: Some Applications of Trigonometry, Chapter 9
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Explanation
- Key concept: Angle of depression from the top equals the angle of elevation from the ship's position (alternate interior angles with the horizontal). Use tan ratio in each right triangle.
- Common mistake: Forgetting that the ships are on opposite sides, so their distances add up (not subtract).
- Examiner expects: Diagram/setup (1 mark), two tan equations (2 marks), sum = given distance (1 mark), final answer (1 mark). Always state the conclusion clearly.