A spherical balloon of radius r subtends an angle . . .

Question : A spherical balloon of radius r subtends an angle θ at the eye of an observer. If the angle of elevation of its centre is Φ, find the height of the centre of the balloon.

Doubt by Saumya

Solution : 

Let the height of the centre of the balloon = h
Radius of the balloon = OP = r (Given)

OA = d

OP丄 AP [Tangent at any point of the circle is perpendicular to the radius through the point of contact]
∠OAP = ∠OAQ = θ/2 (By Symmetry)

In Rt. ΔOAP
sinθ/2 = P/H
sinθ/2 = r/d
d = r/sinθ/2
d = rcosecθ/2 [∵cosecθ=1/sinθ]

d = rcosecθ/2 — (1)

Now, In Rt. ΔOBA
sinΦ = h/d
h = dsinΦ
h = (rcosecθ/2)sinΦ
h = rsinΦcosecθ/2

Hence, the height of the centre of the balloon is rsinΦcosecθ/2.