Aditya is standing on a light house of 150 m height and watching two boats on either side with angles of depression of 30^o and 60^o, respectively. Find the distance between the two boats.

Correct option is D

Given:

Height of lighthouse = 150 m

Angles of depression to two boats = $$30^\circ$$​ and $$60^\circ$$​​

Boats are on opposite sides

Concept Used:

Angle of depression = angle of elevation

​$$\tan\theta = \frac{\text{height}}{\text{perpendicular}}$$​​

Solution:

Distance of boat 1 (angle $$30^\circ$$​):

​$$d_1 = \frac{150}{\tan 30^\circ}$$​​

​$$\tan 30^\circ = \frac{1}{\sqrt{3}}$$​​

​$$d_1 = 150\sqrt{3}$$​​

Distance of boat 2 (angle $$60^\circ$$​):

​$$d_2 = \frac{150}{\tan 60^\circ}$$​​

​$$\tan 60^\circ = \sqrt{3}$$​​

​$$d_2 = \frac{150}{\sqrt{3}} = 50\sqrt{3}$$​​

Total distance between the boats (opposite sides):

D = $$150\sqrt{3} + 50\sqrt{3} = 200\sqrt{3}\text{ m}$$​​

Alternate Solution(Exam Hall Method):