Question

As observed from the top of a $80m$ tall lighthouse, the angles depression of two ships on the same side of the light house in horizontal line with its base are ${30}^{\circ }$ and ${60}^{\circ }$ respectively. Find the distance between the two ships. Give your answer correct to the nearest metre.

Answer 1

Height of tower $80$ m.

Two ships are at $B$ and $C$.

Following the given figure, we can say

$\tan {60}^{\circ }=\frac{AD}{AC}$
$\Rightarrow \sqrt{3}=\frac{80}{AC}$

$\Rightarrow AC=\frac{80}{\sqrt{3}}$

$\tan {30}^{\circ }=\frac{AD}{AB}$
$\Rightarrow \frac{1}{\sqrt{3}}=\frac{80}{AB}$

$\Rightarrow AB=80\times \sqrt{3}$

$\Rightarrow AC+BC=80\sqrt{3}$

$\Rightarrow \frac{80}{\sqrt{3}}+BC=80\sqrt{3}$

$\Rightarrow BC=80\sqrt{3}-\frac{80}{\sqrt{3}}$

$BC=\frac{160}{\sqrt{3}}$

$BC=92.37m$

Nearest integer $=92m$