Question

An observed from the top of a 150 m tall light house, the angles of depression of two ships approaching it are 30° and 45°. If one ship is directly behind the other, find the distance between the two ships.

Answer 1

Let AB be the lighthouse of 150 m and angle of depression of two ship C and D are 30° and 45° respectively.

Let BC = x, CD = y and ∠ADB = 30°, ∠ACB = 45°

We use trigonometric ratios.

In a triangle ABC

⇒ `tan 45^circ = (AB)/(BC)`

⇒ `1 = 150/x`

⇒ x = 150

Again in a triangle ABD

⇒ `tan 30^circ = (AB)/(BD)`

⇒ `1/sqrt3 = 150/(x + y)`

⇒ `x + y = 150sqrt(3)`

⇒ `150 + y = 150sqrt(3)`

⇒ `y = 150sqrt(3) - 150`

⇒ `y = 150(sqrt(3) - 1)`

⇒ y = 150 × 0.732

Hence, distance between the ships is 109.8 m.