Question

The angles of depression of two ships from the top of a light house and on the same side of it are found to be ${45}^0$ and ${30}^0$ respectively. If the ships are $200$ m apart, find the height of the light house.

Answer 1

Let Height of the Light House $=AB=h$

Let the Position of the Ships be $C$ and $D$

and it is Given $CD=200m$

Now, in $△ABC$

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

and in $△ABD$

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

Now, $CD=BD-BC=200\Rightarrow h\sqrt{3}-h=200$

$\Rightarrow$ $h=\frac{200}{\sqrt{3}-1}=\frac{200}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}=100(\sqrt{3}+1)$

$\therefore$ Height of the light house is $100(\sqrt{3}+1)m$