Question

Two eagles are flying in the sky at a height of `10sqrt3` m above the ground. A boy standing at a point on the ground looks towards the eagles, making angles of elevation of 60° and 30° respectively. Find the distance between these two eagles.

Answer 1

Given:

The height of both eagles (h): `10sqrt3`

Angle of elevation of the 1st eagle (θ₁): 60°

Angle of elevation of the 2nd eagle (θ₂): 30°

Let x₁ be the horizontal distance of the first eagle from the boy, and x₂ be the distance of the second eagle.

⇒ For the first eagle (60°):

`tan(60°) = ("Height")/("Distance" (x_1))`

`sqrt3 = (10sqrt3)/x_1`

`x_1 = (10sqrt3)/sqrt3`

∴ x₁ = 10 m

⇒ For the second eagle (30°):

`tan(30°) = ("Height")/("Distance" (x_2))`

`1/sqrt3 = (10sqrt3)/x_2`

x₂ = `10sqrt3 xx sqrt3`

x₂ = 10 × 3

∴ x₂ = 30 m

⇒ Case 1: Eagles are on the same side of the boy:

Distance = x₂ − x₁

= 30 m − 10 m

∴ Distance = 20 m

If eagles are on the same side of the boy, the distance will be 20 m.

⇒ Case 2: Eagles are on the opposite side of the boy:

Distance = x₂ + x₁

= 30 m + 10 m

∴ Distance = 40 m

If eagles are on the opposite side of the boy, the distance will be 40 m.