Question

An observed from the top of a 150 m tall lighthouse, 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^@ = (AB)/(BC)`

`=> 1 = 150/x`

`=> x = 150`

Again in a triangle ABD

`=> tan 30° = (AB)/(BD)`

`=> 1/sqrt3 = 150/(x + y)`

`=> x + y = 150sqrt3`

`=>  150 + y = 150sqrt3`

`=> y = 150sqrt3 - 150`

`=> y = 150(sqrt3 - 1)`

`=> y = 150 xx 0.732`

Hence distance between the ships is 109.8 m