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.

$Spherical balloon with observer point B, angles 60 and 45 degrees$

Question

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.

$Spherical balloon with observer point B, angles 60 and 45 degrees$

[IMAGE_1]

Accepted Answer

Step 1: Let Point $B$ represents observer.
$	herefore \angle QBP = 60°$; $\angle ABO = 45°$
Using geometry $\angle PBO = \frac{1}{2} 	imes 60° = 30°$
Step 2: Now, $\frac{r}{OB} = \sin 30° = \frac{1}{2} \Rightarrow OB = 2r$ ... (i)
Step 3: Also $\frac{OA}{OB} = \sin 45° = \frac{1}{\sqrt{2}} \Rightarrow OB = OA\sqrt{2}$ ... (ii)
Step 4: Using (i) and (ii): $OA = \sqrt{2} \cdot r$
or height of centre of balloon $= \sqrt{2} \cdot r$ units

Question

Solution