Question

A round balloon of radius 'a' subtends an angle θ at the eye of the observer while the angle of elevation of its centre is Φ. Prove that the height of the centre of the balloon is a sin Φ cosec `θ/2`.

Answer 1

Let C be the centre of the balloon, O be the position of man's eye.
Let h be the height of the centre of the balloon.
Then, ∠AOB = θ
So, ∠BOC = ∠COA = `θ/2`

In ΔOAC,
sin θ/2 = `a/"OC"`

⇒ OC = a cosec `θ/2`

In ΔCOD,

`sin Φ = h/"OC"`

`⇒ h = OC sin Φ`

`⇒ h = a cosec θ/2 sin Φ`

`⇒ h = a sin Φ cosec θ/2`