Question

A round balloon of radius $r$ subtends an angle $\alpha$at the eye of the observer, while the angle of elevation of its center is $\beta$. The height of the center of the balloon is

• A. $r\cos ec\alpha \sin \frac{\beta }{2}$
• B. $r\sin \alpha \cos ec\frac{\beta }{2}$
• C. $r\sin \frac{\alpha }{2}\cos ec\beta$
• D. $r\cos ec\frac{\alpha }{2}\sin \beta$

Answer 1

The correct option is D

$r\cos ec\frac{\alpha }{2}\sin \beta$

Explanation for the correct option:

Step 1 : Interpretation from given information

Let $O$ be the center of the balloon and $P$ be the eye of the observer

And $\angle APB$ is the angle subtended by the balloon

Therefore, $\angle APB=\alpha$

and $\angle APO=\angle BPO$

$=\frac{\alpha }{2}$

Step 2 : Find distance $OP$

In $△OAP$, $\sin \frac{\alpha }{2}=\frac{OA}{OP}$

$\Rightarrow$ $\sin \frac{\alpha }{2}=\frac{r}{OP}$

$\Rightarrow$ $OP=r\cos ec\frac{\alpha }{2}$

Step 3 : Find height of the centre of a balloon

In $△OPC$, $\sin \beta =\frac{OL}{OP}$

$\Rightarrow$ $OL=OP\sin \beta$

Therefore, $OL=rcosec\frac{\mathrm{\alpha }}{2}sin\beta$

Thus, the height of the balloon is $\mathit{r}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{e}\mathit{c}\frac{\mathbf{\alpha }}{\mathbf{2}}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\beta }$

Hence, Option ‘D’ is Correct.