A ladder is placed against a wall such that it just reaches the top of the wall. The foot of the ladder is at a distance of 5 metres from the wall. The angle of elevation of the top of the wall from the base of the ladder is 15°. What is the length (in metres) of the ladder?

Solution (By Examveda Team)

$$\eqalign{ & \cos {15^ \circ } = \frac{{BC}}{{AC}} \cr & \frac{{\sqrt 3 + 1}}{{2\sqrt 2 }} = \frac{5}{{AC}} \cr & \frac{{\sqrt 6 + \sqrt 2 }}{{2 \times 2}} = \frac{5}{{AC}} \cr & \therefore AC = \frac{{20}}{{\sqrt 6 + \sqrt 2 }} \cr & AC = \frac{{20\left( {\sqrt 6 - \sqrt 2 } \right)}}{{\left( {\sqrt 6 + \sqrt 2 } \right)\left( {\sqrt 6 - \sqrt 2 } \right)}} \cr & AC = \frac{{20}}{4}\left( {\sqrt 6 - \sqrt 2 } \right) \cr & AC = 5\sqrt 6 - 5\sqrt 2 \cr} $$