Question

A spherical balloon of radius r subtends an angle theta at the eye of an observer. If the angle of elevation of
its centre is alpha, find the height of the centre of the balloon.
​

Answer 1

Answer

Let O be the center of the balloon of radius r and P be the eye of an observer

Let PA and PB be the tangents from P to the balloon.

Then, ∠APX=α

∠APO=∠BPO=

2

α

Let OL be the tar drawn from center O on PX

∠OPL=β

In △OAP

sin

2

α

=

OP

OA

=

OP

r

OP=rcsc

2

α

In △OPL,sinβ=

OP

OL

OL=OPsinβ=rcsc

2

α

.sinβ

h=rcsc

2

α

sinβ