Question

A man in a boat rowing away from a lighthouse 150 m high, takes 2 minutes to change the angle of elevation of the top of the lighthouse from 60° to 45°. Find the speed of the boat.

Answer 1

Let AB be the lighthouse and C and D be the two positions of the boat such that AB = 150 m, ∠ADB = 45° and ∠ACB = 60°.

Let speed of the boat be x metre per minute. 

Therefore, CD = 2x m; 

In ΔADB, 

`(AB)/(DB) = tan 45^circ`

`=>` BD = 150 m

In ΔABC,

`(AB)/(BC) = tan 60^circ = sqrt(3)`

`=> 150/(BC)= sqrt(3)`

`=> BC = 150/ sqrt(3)`

= `150/1.732`

= 86.605 m

∴ CD = BD – BC

= 150 – 86.605

= 63.395 m

`=>` 2x = 63.395

`=> x = 63.395/2`

= 31.6975 m/min

= `31.6975/60` m/sec

= 0.53 m/sec

Hence, the speed of the boat is 0.53 m/sec