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.


A spherical balloon

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 = \frac{1}{2} ∠QBP = 30°

Now, let’s look at ΔOPB

\frac{OP}{OB}  = sin 30°

\therefore         \frac{r}{OB} = \frac{1}{2}

OB = 2r

Now, let’s look at ΔOBA

\frac{OA}{OB}   = Sin 45°

\therefore       \frac{OA}{2r}  =  \frac{1}{\sqrt2}

OA = r√2

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

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