P and Q are two points on the ground on either side of a pole. The angles of elevation of the top of the pole as observed from P and Q are 60° and 30°, respectively and the distance between them is 84√3 m. What is the height (in m) of the pole?

Solution (By Examveda Team)

\[{60^ \circ } = \frac{{\sqrt 3 }}{1}\begin{array}{*{20}{c}} \to &h \\ \to &{\frac{h}{{\sqrt 3 }}} \end{array}\]
$$PN = \frac{h}{{\sqrt 3 }}$$
\[{30^ \circ } = \frac{1}{{\sqrt 3 }}\begin{array}{*{20}{c}} \to &h \\ \to &{\sqrt 3 h} \end{array}\]
$$\eqalign{ & NQ = \sqrt 3 h \cr & PN + NQ = 84\sqrt 3 \cr & \frac{h}{{\sqrt 3 }} + \sqrt 3 h = 84\sqrt 3 \cr & h\left( {1 + 3} \right) = 84 \times \sqrt 3 \times \sqrt 3 \cr & h = \frac{{84 \times 3}}{4} \cr & h = 63\,{\text{m}} \cr} $$