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

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MarkFL

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Feb 24, 2012
13,775
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.
 
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MarkFL

Administrator
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Feb 24, 2012
13,775
Hello Darshan Amin,

Please consider the following diagram:

darshan.jpg

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.