question_answer

From an aeroplane flying, vertically above a horizontal road, the angles of depression of two consecutive stones on the same side of the aeroplane are observed to be \[{{30}^{o}}\] and \[{{60}^{o}}\] respectively. The height at which the aeroplane is flying in km is :

A)  \[\frac{4}{\sqrt{3}}\]                                    

B)  \[\frac{\sqrt{3}}{2}\]

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

D)  2

Correct Answer: B

Solution :

Let the distance of two consecutive stones are x,  \[x+1\].                               In \[\Delta BCD,\]                                 \[\tan \,{{60}^{o}}=\frac{h}{x}\] \[\Rightarrow \]               \[x=\frac{h}{\sqrt{3}}\]                 ??.(i) In  \[\Delta ABC,\]                 \[\tan {{30}^{o}}=\frac{h}{x+1}\] \[\Rightarrow \]               \[\frac{1}{\sqrt{3}}=\frac{h}{x+1}\] \[\Rightarrow \]               \[\frac{h}{\sqrt{3}}+1=\sqrt{3}\,h\]    (From (i)) \[\Rightarrow \]               \[\frac{2h}{\sqrt{3}}=1\] \[\Rightarrow \]               \[h=\frac{\sqrt{3}}{2}km\]