Question

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 ${60}^{\circ }$ and from the same point the angle of elevation of the top of the pedestal is ${45}^{\circ }$. Find the height of the pedestal.

Answer 1

Let $AB$ be the length of the statue and $BC$ be the length of the pedestal.

In $△BCD$,

$\frac{BC}{CD}=\tan {45}^o$

$\therefore BC=CD$

In $△ACD$,

$\tan {60}^o=\frac{AC}{CD}$

$\frac{AB+BC}{CD}=\sqrt{3}$

$BC+1.6=\sqrt{3}CD$

$BC+1.6=\sqrt{3}BC$

$BC=\frac{1.6}{\sqrt{3}-1}$

$BC=\frac{1.6(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)}$

$=\frac{1.6(\sqrt{3}+1)}{(\sqrt{3}{)}^2-(1{)}^2}=\frac{1.6(\sqrt{3}+1)}{2}$

$=0.8(\sqrt{3}+1)$ m