Question

A man in a boat rowing away from a lighthouse 100m high takes 2 minutes to change angle of elevation of the top of a lighthouse from 60 degree to 30 degree find the speed of boat in metres per minute.

Answer 1

Answer is in the attachment. Hope it is helpful!

Answer 2

Given,

The height of the lighthouse = 100m

The time taken = 2 mins

Initial angle of elevation = 60°

Final angle of elevation = 30°

To find,

Speed of boat in m/min

Solution,

let the height of the building be AB = 100m,

the distance from the base when the elevation angle was 60° be BC= x

and distance traveled (when the angle changes to 30°) be CD = y

In ΔABC

tan60° = AB/BC

$\sqrt{3}$ = 100/x

⇒ x = 100/$\sqrt{3}$ × $\sqrt{3}$/$\sqrt{3}$

⇒ x = 100$\sqrt{3}$/3 ________ (1)

In  ΔADB

tan30° = AB/BD

⇒ 1/$\sqrt{3}$ = 100/(x+y)

⇒ x+y = 100$\sqrt{3}$

⇒ y = 100$\sqrt{3}$ - (100$\sqrt{3}$)/3

⇒ y = 100$\sqrt{3}$ (1 - 1/3)

⇒ y = 100$\sqrt{3}$ × 2/3

⇒ y = 200 × 1.73/3

⇒ y = 115.33 m

now,

speed = distance / time

           = 115.33/2

           = 57.66 m/min

∴ The speed of the boat is 57.66 m/min.