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 ${30}^{\circ }$ and ${45}^{\circ }$ respectively. If the lighthouse is 100 m high, the distance between the two ships is:

• A. 173 m
• B. 200m
• C. 300m
• D. 273 m

Answer 1

The correct option is D

273 m

Let AB be the lighthouse and C and D be the positions of the ships.


Then, AB = 100 m, $\angle ACB={30}^{\circ }and\angle ADB{45}^{\circ }$
$\frac{AB}{AC}=tan{30}^{\circ }\frac{1}{\sqrt{3}}\Rightarrow AC=AB\times \sqrt{3}=100\times \sqrt{3}m\frac{AB}{AD}=tan{45}^{\circ }=1\Rightarrow AD=AB=100mSo,CD=(AC+AD)=(100\sqrt{3}+100)=100(\sqrt{3}+1)=(100\times 2.73)=273m$