Question

# Two boats are sailing in the sea on either side of a lighthouse. At a particular time the angles of depression of the two boats, as observed from the top of the lighthouse are 45∘ and 30∘ respectively. If the lighthouse is 100 m high, find the distance between the two boats.

A
273.2 m
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B
172.4 m
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C
380 m
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D
220.2 m
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Solution

## The correct option is B 273.2 mLet A and B be the position of two boats (figure)Let PM be the lighthouse such that PM =100 mLet AM =x and BM =yIn ΔAPM,AMPM=cot30∘⇒x100=√3⇒x=100√3 m [∵ cot30o=√3]Similarly, in Δ BPM,BMPM=cot45∘⇒y100=1⇒y=100 m [∵ cot45o=1]∴ Required distance, AB=x+y⇒100√3+100=100(√3+1)m=273.2 mHence, the distance between the two boats is 273.2 m

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