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A spherical balloon of radius r subtends angle \( \Large 60 ^{\circ} \) at the eye of an observer. If the angle of elevation of its centre is \( \Large 60 ^{\circ} \) and h is the height of the centre of the balloon, then which one of the following is correct?

A) h=r
B) \( \Large h = \sqrt{2}r \)
C) \( \Large h = \sqrt{3}r \)
D) h=2r

Correct Answer:



C) \( \Large h = \sqrt{3}r \)

Description for Correct answer:

In \( \Large \triangle ABO \),

\( \Large \sin 60 ^{\circ} = \frac{OB}{AO} \)

=> \( \Large AO = \frac{OB}{\sin 60 ^{\circ} } \)

Now, in \( \Large \triangle AOC, \)

\( \Large \sin \frac{60 ^{\circ} }{2} = \frac{OC}{AO} \)

=> \( \Large AO = \frac{OC}{\sin 30 ^{\circ} } \)

From Eqs. (i) and (ii), we get

\( \Large \frac{OB}{\sin 60 ^{\circ} }=\frac{OC}{\sin 30 ^{\circ} }3 \)

=> \( \Large \frac{h}{\frac{\sqrt{3}}{2}}=\frac{r}{\frac{1}{2}} \)

Therefore, \( \Large h = \sqrt{3}r \)

Part of solved Height and Distance questions and answers : >> Elementary Mathematics >> Height and Distance

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Similar Questions

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A). \( \Large - \alpha ,\ - \beta \)

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