Q) A spherical balloon of radius r subtends an angle of 60° at the eye of an observer. If the angle of elevation of its centre is 45° from the same point, then prove that height of the centre of the balloon is √2 times its radius.

Ans: 

Let’s start from drawing the above image. Our observer is at point B.

Its  given that ∠QBP = 60° and ∠ OBA = 45°

Using SAS identity, we can say that ΔOPB and ΔOQB are identical.

Hence ∠OBP = ∠OBQ = ∠QBP = 30°

Now, let’s look at ΔOPB

 = sin 30°

=

OB = 2r

Now, let’s look at ΔOBA

  = Sin 45°

  = 

OA = r√2

Hence, the height of the center of balloon is √2 times of its radius.