Question

As observed from the top of a $150$ m tall lighthouse, the angles of depression of two ships approaching it are ${30}^{\circ }$ and ${45}^{\circ }$. If one ship is directly behind the other, find the distance between the two ships.

Answer 1

Let $AB$ be the lighthouse and the two ships be at point $C$ and $D$ respectively.

$\Rightarrow$ In $△ABC,\tan {45}^o=\frac{AB}{BC}$

$\Rightarrow$ $1=\frac{150}{BC}$

$\therefore$ $BC=150m$

$\Rightarrow$ In $△ABD,\tan {30}^o=\frac{AB}{BD}$

$\Rightarrow$ $\frac{1}{\sqrt{3}}=\frac{150}{150+CD}$

$\Rightarrow$ $150\sqrt{3}=150+CD$

$\Rightarrow$ $CD=150(\sqrt{3}-1)m=109.5m$ $(Take\sqrt{3}=1.73)$

$\therefore$ The distance between two ships is $109.5m$