# Darshan Amin's question at Yahoo! Answers regarding finding the height of a balloon

#### MarkFL

##### Administrator
Staff member
Here is the question:

A round balloon of radius r subtends an angle α at the eye of observer while the angle of elevation of its?

A round balloon of radius r subtends an angle α at the eye of observer while the angle of elevation of its centre is β.prove that the height of the centre of the balloon is r*sinβ*cosecα*1/2
I have posted a link there to this thread so the OP can view my work.

#### MarkFL

##### Administrator
Staff member
Hello Darshan Amin,

Please consider the following diagram: From this we see:

(1) $$\displaystyle \sin(\beta)=\frac{h}{d}\implies h=d\sin(\beta)$$

(2) $$\displaystyle \sin\left(\frac{\alpha}{2} \right)=\frac{r}{d}\implies d=r\csc\left(\frac{\alpha}{2} \right)$$

Substituting for $d$ from (2) into (1) we obtain:

$$\displaystyle h=r\csc\left(\frac{\alpha}{2} \right)\sin(\beta)=r\sin(\beta) \csc\left(\frac{\alpha}{2} \right)$$

Shown as desired.