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)  \[\frac{10}{\sqrt{3}}\,\,\text{m}\]

B)  \[10\sqrt{3}\]

C)  \[\frac{30}{\sqrt{3}}\,\,\text{m}\]

D)  \[30\,\,\text{m}\]

Correct Answer: D

Solution :

Let RQ be the height of building, then RQ = 10 m, S be    the position    of helicopter, then In\[\Delta PQR,\]

\[\frac{RQ}{PQ}=\tan 30{}^\circ \]

\[\Rightarrow \]   \[PQ=\frac{RQ}{\tan 30{}^\circ }=10\sqrt{3}\]

In \[\Delta SPQ,\]

\[\tan 60{}^\circ =\frac{SQ}{PQ}\Rightarrow \frac{SQ}{PQ}=\sqrt{3}\]

\[SQ=PQ\times \sqrt{3}\]

\[=10\sqrt{3}\times \sqrt{3}=30\,\,\text{m}\]