Question

From the top of a $50m$ high tower, the angles of depression of the top and bottom of a pole are observed to be ${45}^o$ and ${60}^o$ respectively. Find the height of the pole.

Answer 1

Let the height of the pole be $h$ and the distance between the pole and tower be $x$.

It can be observed that,

$\tan {60}^{\circ }=\frac{50}{x}$

$\sqrt{3}=\frac{50}{x}$

$x=\frac{50}{\sqrt{3}}$

And,

$\tan {45}^{\circ }=\frac{50-h}{x}$

$1=\frac{50-h}{x}$

$50-h=x$

$50-h=\frac{50}{\sqrt{3}}$

$h=50-\frac{50}{\sqrt{3}}$

$h=\frac{50\left(\sqrt{3}-1\right)}{\sqrt{3}}$

$h=21.1m$

Therefore, the height of the pole is $21.1m$.