Question

Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are ${30}^o$ and ${45}^o$ respectively. If the lighthouse is $100\text{m}$ high, find the distance between the two ships.

Answer 1

Let $AB$ be the light house, $C$ and $D$ be the ships respectively

Let $AC=x$ and $AD=y$

In $△ABC$

$\begin{array}{c}\\ \tan {30}^{\circ }=\frac{AB}{AC}\\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{100}{x}\\ \Rightarrow x=100\sqrt{3}\end{array}$

similarly,

In $△ABD$

$\begin{array}{c}\\ \tan {45}^{\circ }=\frac{AB}{AD}\\ \Rightarrow 1=\frac{100}{y}\\ \Rightarrow y=100\end{array}$

Hence,

Distance between two ships $=\left(x+y\right)$

$\begin{array}{c}=100\sqrt{3}+100\\ =100\left(\sqrt{3}+1\right)\text{m}\end{array}$