Two ships are sailing in the sea on the top sides of a lighthouse.The angle of a elevation of the top of the lighthouse is observed from the ships are 30? and 45? respectively. If the lighthouse is 100 m heigh, find the distance between two ships?

Let AB be the lighthouse and C and D be the positions of the ships.
\begin{aligned}
ext{AB = 100m}, \angle{ACB}=30^{\circ}, \\
\angle{ADB}=45^{\circ}\\
\frac{AB}{AC} = tan&30{\circ} = \frac{1}{\sqrt{3}} \\
=> AC = AB*\sqrt{3} = 100\sqrt{3}m\\

\frac{AB}{AD} = tan&45^{\circ} = 1 \\
=> AB = AD = 100m\\

CD = AC+AD\\
= (100\sqrt{3}+100)&m \\
= 100(\sqrt{3}+1)&m\\
= 100*2.73&m
= 273m