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From a light-house, the angles of depression of two ships on opposite sides of the light-house are $30 °$ and $45 °$ . If the height of the light-house is 100 m, then what will be the distance between the ships if the line joining them passes through the foot of the light-house?

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273.2m

237.2m

248m

247.9m

answer is A.

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Detailed Solution

Given that,
The height of the light-house

A B = 100

m,
We draw a diagram,
Question Image

The two ships be at C and D
The angles of depression =

45 °

30 °

We use the trigonometric identity,

tan θ = p e r p e n d i c u l a r b a s e

Applying the tangent rule in

Δ A B D

tan 30 ° = A B B D ⇒ 13 = 100 B D tan 30 ° = 13 ⇒ B D = 1003

Applying the tangent rule in

Δ A B C

tan 45 ° = A B B C ⇒ 1 = 100 B C tan 45 ° = 1 ⇒ B C = 100 m

For the distance between two ships, we add the two values,

d = B D + B C ⇒ d =100 3 + 100 ⇒ d = 100 1 + 1.732 ⇒ d = 273.2 m

Thus the distance between the ships will be 273.2 m.
Therefore the correct option is 1.

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