question_answer

A flagstaff 5m high is placed on a building 25m high. If the flag and building both subtend equal angles on the observer at a height 30m, the distance between the observer and the top of the flag is

A)  \[\frac{5\sqrt{3}}{2}\]                          

B)  \[5\sqrt{\frac{3}{2}}\]

C)  \[5\sqrt{\frac{2}{3}}\]                          

D)  \[\frac{5\sqrt{2}}{3}\]

Correct Answer: B

Solution :

In \[\Delta OBC,\] we have \[\tan \alpha =\frac{5}{x}\]                                ?(i) \[\tan 2\alpha \,=\frac{30}{x}\]                           ?(ii) Dividing (ii) by (i) we have \[\tan 2\alpha \,=\frac{30}{5}\tan \alpha \] \[\Rightarrow \,\,\frac{2\tan \alpha }{1-{{\tan }^{2}}\alpha }\,=6\tan \alpha \Rightarrow \,{{\tan }^{2}}\alpha \,=\frac{2}{3}\] \[\Rightarrow \,\,\tan \alpha =\sqrt{\frac{2}{3}}\] \[\therefore \,\,x=5\,\cot \,\alpha \,=\sqrt{\frac{3}{2}}\]