question_answer

A flag-staff of 5 m high stands on a building of 25 m high. At an observer at a height of 30 m. The flag-staff and the building subtend equal angles. The distance of the observer from the top of the flag-staff is [EAMCET 1993]

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

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

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

D) None of these

Correct Answer: B

Solution :

We have \[\tan \alpha =\frac{5}{x}\] and \[\tan 2\alpha =\frac{30}{x}\] \[\therefore \] \[\tan 2\alpha =\frac{30}{5\cot \alpha }\Rightarrow \tan 2\alpha =6\tan \alpha \] \[\Rightarrow \] \[3-3{{\tan }^{2}}\alpha =1\Rightarrow \tan \alpha =\sqrt{\frac{2}{3}}\] \[\therefore \] \[x=5\cot \alpha =5\sqrt{\frac{3}{2}}\].