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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.

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Answer

Let AB be the lighthouse and C and D be the position of 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. NCERT Class 10 Mathematics CBSE Solutions.

We know that,

Alternate angles are equal.

From figure,

⇒ ∠ACB = ∠EAC = 45°

⇒ ∠ADB = ∠EAD = 30°

In △ABC,

tan 45° = Side opposite to angle 45°Side adjacent to angle 45°\dfrac{\text{Side opposite to angle 45°}}{\text{Side adjacent to angle 45°}}

Substituting values we get :

⇒1=ABBC⇒BC=AB=75 m.\Rightarrow 1 = \dfrac{AB}{BC} \\[1em] \Rightarrow BC = AB = 75 \text{ m}.

In △ABD,

tan 30° = Side opposite to angle 30°Side adjacent to angle 30°\dfrac{\text{Side opposite to angle 30°}}{\text{Side adjacent to angle 30°}}

Substituting values we get :

⇒13=ABBD⇒BD=AB3=753 m.\Rightarrow \dfrac{1}{\sqrt{3}} = \dfrac{AB}{BD} \\[1em] \Rightarrow BD = AB\sqrt{3} = 75\sqrt{3} \text{ m.} \\[1em]

From figure,

CD = BD - BC = 753−75=75(3−1)75\sqrt{3} - 75 = 75(\sqrt{3} - 1).

Hence, distance between two ships = 75(3−1)75(\sqrt{3} - 1) meters.

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