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. $\dfrac{5}{2}$
B. \[5\sqrt {\dfrac{3}{2}} \]
C. $5\sqrt {\dfrac{2}{3}} $
D. None of these

Answer 1

Hint: As the angle subtended by the building and flag-staff is equal so we can use the tangent property i.e., $\tan \theta $ = (perpendicular/base), keeping the base same and then eliminating the variable which is not required i.e., $\theta $ .

Complete step-by-step answer: In the question, it is given that the height of flag-staff BC is 5 m and the height of building AB is 25 m.

     

Let the distance of the observer from the top of the flag-staff be OC and the angle subtended by the flag-staff \[\angle BOC\] and building \[\angle AOB\] be $\theta $.
Thus, using the tangent property for the \[\vartriangle \]BOC, we get
$\tan \theta $ = (perpendicular/base) = \[\dfrac{{BC}}{{OC}}\] = \[\dfrac{5}{{OC}}\]
For the\[\vartriangle \]AOC, we get
\[\tan 2\theta \] = (perpendicular/base) = \[\dfrac{{AC}}{{OC}}\] = \[\dfrac{{AB + BC}}{{OC}}\] = \[\dfrac{{30}}{{OC}}\]
and, \[\tan 2\theta \] = $\dfrac{{2\tan \theta }}{{1 - {{\tan }^2}\theta }}$ = \[\dfrac{{30}}{{OC}}\]
Therefore, substituting the value of tan $\theta $ in the above formula, we get
\[\tan 2\theta \] = \[\dfrac{{2 \times \dfrac{5}{{OC}}}}{{1 - {{\left( {\dfrac{5}{{OC}}} \right)}^2}}}\] = \[\dfrac{{30}}{{OC}}\]
Or, \[\dfrac{1}{{1 - {{\left( {\dfrac{5}{{OC}}} \right)}^2}}}\] = 3
Or, \[3\left( {O{C^2} - 25} \right)\] = \[3\left( {O{C^2} - 25} \right)\]
Or, \[3\left( {O{C^2} - 25} \right)\] = \[3O{C^2} - 75\]
Or, \[2O{C^2}\] = 75
Or, OC = \[\sqrt {\dfrac{{75}}{2}} \] = \[5\sqrt {\dfrac{3}{2}} \]
Thus, the distance of the observer from the top of the flag-staff is \[5\sqrt {\dfrac{3}{2}} \] m.

So, the correct answer is “Option B”.

Note: There is another approach to the same question. Instead of substituting value from one equation into the other equation, we can also divide the second equation from the first equation which will eliminate OC and give us the value of $\theta $ and then we can calculate OC by using
OC = $\dfrac{5}{{\tan \theta }}$ . This key point helps in solving the questions whenever angle as well as distance are required.