Question:
If the length of the shadow of a tower is $\sqrt{3}$ times its height then the angle of elevation of the sun is
(a) 45°
(b) 30°
(c) 60°
(d) 90°
Solution:
(b) 30°
Let $A B$ be the pole and $B C$ be its shadow.
Let $A B=h$ and $B C=x$ such that $x=\sqrt{3} h$ (given) and $\theta$ be the angle of elevation.
From $\triangle A B C$, we have:
$\frac{A B}{B C}=\tan \theta$
$\Rightarrow \frac{h}{x}=\frac{h}{\sqrt{3} h}=\tan \theta$
$\Rightarrow \tan \theta=\frac{1}{\sqrt{3}}$
$\Rightarrow \theta=30^{\circ}$
Hence, the angle of elevation is $30^{\circ}$.