Question:
As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
Updated On: Jan 13, 2026
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Solution and Explanation
Let AB be the lighthouse and the two ships be at point C and D respectively.
In ∆ABC,
\(\frac{AB}{ BC} = tan 45^{\degree}\)
\(\frac{75}{ BC} = 1\)
\(BC = 75\,m\)
In ∆ABD,
\(\frac{AB}{ BD}= tan 60^{\degree}\)
\(\frac{75}{ BC +CD} = \frac{1}{\sqrt3}\)
\(\frac{75}{ 75 + CD} = \frac1{ \sqrt3}\)
\(75 \sqrt3 = 75 + CD\)
\(75 (\sqrt3 -1)m = CD\)
Therefore, the distance between the two ships is \(75(\sqrt3 -1) \,m\).
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