Question

Two poles of equal heights are standing opposite to each other on either side of the road which is 80 m wide. From a point P between them on the road, the angle of elevation of the top of one pole is ${60}^{\circ }$ and the angle of depression from the top of another pole at P is ${30}^{\circ }$. Find the height of each pole and distances of the point P from the poles.

Answer 1

Let AB and CD be the two poles of equal height and their heights be h m. BC be the 80 m wide road. P be any point on the road.

Let CP be x m, therefore BP = (80 – x) .
Also, ∠APB = 60° and ∠DPC = 30°

In right angled triangle DCP,

$tan30°=\frac{CD}{CP}\Rightarrow \frac{h}{x}=\frac{1}{\sqrt{3}}\Rightarrow h=\frac{x}{\sqrt{3}}----------\left(1\right)$

In right angled triangle ABP,
$tan60°=\frac{AB}{AP}\Rightarrow \frac{h}{(80–x)}=\sqrt{3}\Rightarrow h=\sqrt{3}(80–x)\Rightarrow \frac{x}{\sqrt{3}}=\sqrt{3}(80–x)\Rightarrow x=3(80–x)\Rightarrow x=240–3x\Rightarrow x+3x=240\Rightarrow 4x=240\Rightarrow x=60$

Height of the pole, $h=\frac{x}{\sqrt{3}}=\frac{60}{\sqrt{3}}=20\sqrt{3}$.

Thus, position of the point P is 60 m from C and height of each pole is $20\sqrt{3}$ m.