Question

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.

Answer 1

Let AB be the lighthouse and the two ships be at point C and D respectively.

In ΔABC,

`"AB"/"BC"` = tan 45°

`75/"BC"` = 1

BC = 75 m

In ΔABD,

`"AB"/"BD"` = tan 30°

`75/("BC" + "CD") = 1/sqrt3`

`75/(75+"CD") = 1/sqrt3`

`75sqrt3 = 75 + "CD"`

`75(sqrt3 -1)"m" = "CD"`

If we take the value of `sqrt3` = 1.73

`75(1.73 -1)"m"`

= 54.91 m.

Hence, the Distance between the two ships is 54.91 m.