To a man standing outside his house, the angles of elevation of the top and bottom of a window are 60° and 45° respectively. If the height of the man is 180 cm and he is 5 m away from the wall, what is the length of the window?

Answer: Option D

Let AB be the man and CD be the window
Given that the height of the man, AB = 180 cm, the distance between the man and the wall, BE = 5 m,
∠ DAF = 45°, ∠ CAF = 60°
From the diagram, AF = BE = 5 m
$$\begin{array}{rl}& \text{From}\text{the}\text{right}\Delta AFD,\\ & \tan 45^{\circ }=\frac{DF}{AF}\\ & 1=\frac{DF}{5}\\ & DF=5......\left(1\right)\\ & \text{From}\text{the}\text{right}\Delta AFC,\\ & \tan 60^{\circ }=\frac{CF}{AF}\\ & \sqrt{3}=\frac{CF}{5}\\ & CF=5\sqrt{3}......\left(2\right)\\ & \text{Length}\text{of}\text{the}\text{window}\\ & =CD=\left(CF-DF\right)\end{array}$$
  $$=5\sqrt{3}-5$$     [∵ Substituted the value of CF and DF from (1) and (2)]
$$\begin{array}{rl}& =5\left(\sqrt{3}-1\right)\\ & =5\left(1.73-1\right)\\ & =5\times 0.73\\ & =3.65\text{m}\end{array}$$