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.
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.