Question

Two boats approach a light house in mid-sea from opposite directions. The angles of elevation of the light house from two boats are ${30}^o$ and ${45}^o$ respectively. If the distance between two boats is $100$m, find the height of the light house.

Answer 1

According to Question and relating with the diagram

$AB=h\left(Let\right)BD-BC=CD=100$

Now, in $△ABC$

$\tan {45}^o=\frac{h}{BC}\Rightarrow h=BC$

and in $△ABD$

$\tan {30}^o=\frac{h}{BD}\Rightarrow BD=h\sqrt{3}$

and we know that,

$CD=BD-BC=h\sqrt{3}-h=h(\sqrt{3}-1)=100\Rightarrow h=\frac{100}{\sqrt{3}-1}\ast \frac{\sqrt{3}+1}{\sqrt{3}+1}=50(\sqrt{3}+1)$