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A and B are standing on ground 50 meters apart. The angles of elevation for these two to the top of a tree are 60° and 30°. What is height of the tree?

a) $$50\sqrt 3 \,{\text{m}}$$
b) $$\frac{{25}}{{\sqrt 3 }}\,{\text{m}}$$
c) $$25\sqrt 3 \,{\text{m}}$$
d) $$\frac{{25}}{{\sqrt 3 – 1}}\,{\text{m}}$$

Answer: c
Explanation:
$$\eqalign{ & {\text{In}}\,\Delta PBQ,\,\tan {60^ \circ } = \frac{{PQ}}{{BQ}} \cr & BQ = \frac{{PQ}}{{\sqrt 3 }} \cr & {\text{In}}\,\Delta PAQ,\,\tan {30^ \circ } = \frac{{PQ}}{{AQ}} \cr & \frac{1}{{\sqrt 3 }} = \frac{{PQ}}{{50 + BQ}} \cr & PQ = \frac{{50 + BQ}}{{\sqrt 3 }} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{50 + \frac{{PQ}}{{\sqrt 3 }}}}{{\sqrt 3 }} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{50\sqrt 3 + PQ}}{{\sqrt 3 \times \sqrt 3 }} \cr & 3PQ = 50\sqrt 3 + PQ \cr} $$
$$ PQ = 25\sqrt 3 \,{\text{m}} = $$     Height of tree

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