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From a light house, the angles of depression of two ships on opposite sides of the lighthouse are observed to be 30° and 45°. If the height of the lighthouse is h. What is the distance between the ships?

(a) (√3 + 1)h

(b) (√3 - 1) h

(c) √3 h

(d) \(\bigg(1+\frac{1}{\sqrt3}\bigg)\)h

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(a) (√3 + 1)h

Let PQ be the light house whose height = h metre. Let A and B be the position of the ships on opposite sides of the lighthouse such that angle of depression for A and B are 30° and 45° respectively.

Let AQ = x metre, QB = y metres. ∠PAQ = 30°, ∠PBQ = 45°. 

Required distance between the ships = AB = AQ + QB = x + y 

In rt. Δ PAQ, 

tan 30° = \(\frac{h}{x}\) ⇒ \(\frac{1}{\sqrt3}\) = \(\frac{h}{x}\) ⇒ \(x\) = h√3

In rt. Δ PBQ, 

tan 45° = \(\frac{h}{y}\) ⇒ 1 = \(\frac{h}{y}\) ⇒ y = h

∴ x + y = h√3 + h = h(√3 + 1)m.

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