question_answer

At a point P on the ground, the angle of elevation of the top of a 10m tall building and of a helicopter hovering some distance over the top of the building are \[30{}^\circ \] and \[60{}^\circ \] respectively. Then, the height of the helicopter above the ground is

A) \[\frac{10}{\sqrt{3}}\]m                        

B) \[10\sqrt{3}\] m

C) \[\frac{20}{\sqrt{3}}\] m                       

D) 30 m

Correct Answer: D

Solution :

Let RQ be the height of building, then RQ = 10 m. S be the position of helicopter. Then In \[\Delta \] PQR, \[\frac{RQ}{PQ}=\tan 30{}^\circ \] \[\Rightarrow \,\,\,PQ=\frac{RQ}{\tan \,30{}^\circ }=10\sqrt{3}\] \[\therefore \] In \[\Delta \,\,\,SPQ,\,\,\tan \,60{}^\circ =\frac{SQ}{PQ}\] \[\Rightarrow \,\,\frac{SQ}{PQ}=\sqrt{3}\] \[\Rightarrow \,\,SQ=PQ\times \sqrt{3}\] \[\Rightarrow 10\sqrt{3}\times \sqrt{3}=30\] m