A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

Let BC be the pedestal and CD the statue.

In △ABC,

tan 45° = $\dfrac{\text{Side opposite to angle 45°}}{\text{Side adjacent to angle 45°}}$

Substituting values we get :

$\Rightarrow 1 = \dfrac{BC}{AB} \\[1em] \Rightarrow AB = BC = x \text{ (let)}.$

In △ABD,

tan 60° = $\dfrac{\text{Side opposite to angle 60°}}{\text{Side adjacent to angle 60°}}$

Substituting values we get :

$\Rightarrow \sqrt{3} = \dfrac{BD}{AB} \\[1em] \Rightarrow BD = AB\sqrt{3} = x\sqrt{3} \text{ meters}.$

From figure,

$\Rightarrow CD = BD - BC \\[1em] \Rightarrow 1.6 = x\sqrt{3} - x \\[1em] \Rightarrow x(\sqrt{3} - 1) = 1.6 \\[1em] \Rightarrow x = \dfrac{1.6}{\sqrt{3} - 1}$

Rationalising x, we get :

$\Rightarrow x = \dfrac{1.6}{\sqrt{3} - 1} \times \dfrac{\sqrt{3} + 1}{\sqrt{3} + 1} \\[1em] \Rightarrow x = \dfrac{1.6(\sqrt{3} + 1)}{3 - 1} \\[1em] \Rightarrow x = \dfrac{1.6(\sqrt{3} + 1)}{2} \\[1em] \Rightarrow x = 0.8(\sqrt{3} + 1) \text{ meters}.$

Hence, height of pedestal = $0.8(\sqrt{3} + 1)$ meters.