Question

A flag-staff 5 m high stands on a building 25 m high. To 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

• A. $\frac{5}{2}$
• B. $5\sqrt{\frac{3}{2}}$
• C. $5\sqrt{\frac{2}{3}}$
• D. none of these

Answer 1

The correct option is B $5\sqrt{\frac{3}{2}}$

AB = 25 m is building, BC = 5 m is flag-staff. O is an observer at a height of 30 m. OC = x is the distance of the observer from the top of flag-staff AB and BC subtend the same angle at O.
tan $\theta$ = 5/x, tan 2$\theta$ = 30/x
Eliminate x. $\frac{tan2\theta }{tan\theta }$ = 6
1- $ta{n}^2\theta$ = 1/3
tan $\theta$ = $\sqrt{2/3}$
x = 5 cot $\theta$ = $5\sqrt{\frac{3}{2}}$