As observed from the top of a lighthouse, $75 m$ high 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, then the distance between the two ships is:

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54.9m

55.1m

55.7 m

52.9m

answer is A.

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Detailed Solution

Given,
Height of the lighthouse, h=75m
Angle of depression to one of the ship,

θ_{1} = 30 °

And angle of depression to another ship,

θ_{2} = 45 °

Consider the figure,
Question Image

Consider,

∆ ACB,

tan θ = p e r p e n d i c u l a r B a s e ⇒ tan 30 ° = 75 B C ⇒ B C = 75 tan 30 ° ⇒ B C = 753 m ∵ tan 30 ° = 13

Again, consider,

∆ ADB,

tan θ = p e r p e n d i c u l a r B a s e ⇒ tan 45 ° = 75 B D ⇒ B D = 75 tan 45 ° ⇒ B D = 75 m ∵ tan 45 ° = 1

The distance between the ships,

B C − B D = 753 − 75 ⇒ D C = 129.9 − 75 ⇒ D C = 54.9 m

Therefore, the distance between ships is 54.9m.
Hence, option 1 is correct.

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