Question

From the top of a lighthouse of height $200$ meters, the angles of depression of two ships on either side of it are observed to be ${30}^0$ and ${45}^0$, then the distance between the ships in meters is:

• A. $100(\sqrt{2}+1)$
• B. $100(\sqrt{2}-1)$
• C. $100(\sqrt{3}+1)$
• D. $200(\sqrt{3}+1)$

Answer 1

The correct option is D $200(\sqrt{3}+1)$

Let $d$ be the Distance between the two ships and $b$ be the distance between the tower and the ship at which angle of depression is ${45}^{\circ }$

$\tan {45}^{\circ }=\frac{h}{b}=\frac{200}{b}$ where h is the height of the tower
$\Rightarrow b=200m$

$\tan {30}^{\circ }=\frac{h}{d-b}$$\Rightarrow \frac{1}{\sqrt{3}}=\frac{200}{d-200}$

$\Rightarrow d=200(\sqrt{3}+1)m$