Question

Question 8
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 height of statue.
D is the point on the ground from where the elevation is taken.
Height of pedestal $=BC=AC-AB$
As per question,
In right $\Delta BCD$,

$tan{45}^{\circ }=\frac{BC}{CD}$
$\Rightarrow 1=\frac{BC}{CD}$
$\Rightarrow BC=CD$
Also,
In right $\Delta ACD$,
tan ${60}^{\circ }=\frac{AC}{CD}$
$\Rightarrow \sqrt{3}=\frac{AB+BC}{CD}$
$\Rightarrow \sqrt{3}CD=1.6+BC$
$\Rightarrow \sqrt{3}BC=1.6+BC$
$\Rightarrow \sqrt{3}BC-BC=1.6$
$\Rightarrow BC(\sqrt{3}-1)=1.6$
$\Rightarrow BC=\frac{1.6}{(\sqrt{3}-1)}m$
Rationalising we get,

$\Rightarrow BC=\frac{1.6}{(\sqrt{3}-1)}\frac{\sqrt{3}+1}{\sqrt{3}+1}m$
$\Rightarrow BC=0.8(\sqrt{3}+1)m$
Thus, the height of the pedestal is 0.8$(\sqrt{3}+1)$m.