Question

Question 13
As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are ${30}^{\circ }$ and ${45}^{\circ }$. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Answer 1

Let AB be the lighthouse of height 75 m.
Let C and D be the positions of the ships.
${30}^{\circ }$ and ${45}^{\circ }$ are the angles of depression from the lighthouse.

As per question,
In right $\Delta ABC,$
tan ${45}^{\circ }=\frac{AB}{BC}$
$\Rightarrow 1=\frac{75}{BC}$
$\Rightarrow BC=75m$
Also,
In right $\Delta ABD,$
tan ${30}^{\circ }=\frac{AB}{BD}$
$\Rightarrow \frac{1}{\sqrt{3}}=\frac{75}{BD}$
$\Rightarrow BD=75\sqrt{3}m$
The distance between the two ships $=CD=BD-BC=(75\sqrt{3}-75)m=75(\sqrt{3}-1)m.$