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.

Let AB is 75 m high lighthouse and C and D are the two ships.
In ΔABC,
AB/BC = tan 45° ⇒ 75/BC = 1
⇒ BC = 75
In ΔABD,
AB/BD = tan 30°
⇒ 75/(BC + CD) = 1/√3
⇒ 75/(75 + CD) = 1/√3
⇒ 75 + CD = 75√3
⇒ CD = 75(√3 – 1)
Hence, the distance between the two ships is 75(√3 – 1).