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The angles of depression of two ships A and B as observed from the top of a lighthouse 60 m high are 60° and 45° respectively. If the two ships are on the opposite sides of the lighthouse, find the distance between the two ships. Give your answer correct to nearest whole number.

Heights & Distances

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Answer

Let CD be the lighthouse.

The angles of depression of two ships A and B as observed from the top of a lighthouse 60 m high are 60° and 45° respectively. If the two ships are on the opposite sides of the lighthouse, find the distance between the two ships. Give your answer correct to nearest whole number. Heights and Distances, Concise Mathematics Solutions ICSE Class 10.

As, angle of depression are 60° and 45°.

Since, alternate angles are equal.

∴ ∠DAC = ∠EDA = 60° and ∠DBC = ∠FDB = 45°.

In △ACD,

tan 60°=PerpendicularBase⇒3=CDAC⇒AC=CD3⇒AC=601.732⇒AC=34.64 m.\text{tan 60°} = \dfrac{\text{Perpendicular}}{\text{Base}} \\[1em] \Rightarrow \sqrt{3} = \dfrac{CD}{AC} \\[1em] \Rightarrow AC = \dfrac{CD}{\sqrt{3}} \\[1em] \Rightarrow AC = \dfrac{60}{1.732} \\[1em] \Rightarrow AC = 34.64 \text{ m}.

In △BCD,

tan 45°=PerpendicularBase⇒1=CDBC⇒BC=CD=60 m.\text{tan 45°} = \dfrac{\text{Perpendicular}}{\text{Base}} \\[1em] \Rightarrow 1 = \dfrac{CD}{BC} \\[1em] \Rightarrow BC = CD = 60 \text{ m}.

AB = AC + BC = 34.64 + 60 = 94.64 ≈ 95 m.

Hence, the distance between two ships = 95 m.

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