A vertical pole fixed to the horizontal ground is divided in the ratio 3 : 7 by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground 18 m away from the base of the pole, then the height of the pole (in meters) is :
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Official solution
Let height of pole = 10l
\[\tan \alpha = {{3l} \over {18}} = {l \over 6}\]
\[\tan 2\alpha = {{10l} \over {18}}\]
\[{{2\tan \alpha } \over {1 - {{\tan }^2}\alpha }} = {{10l} \over {18}}\]
use \[\tan \alpha = {l \over 6} \Rightarrow l = \sqrt {{{72} \over 5}} \]
height of pole = \[10l = 12\sqrt {10} \]