Answer:
Each boat are 43.86 miles from the lighthouse.
Step-by-step explanation:
From the diagram, at point A the lighthouse is placed and at B, C two boats are placed.
As the two boats are equidistant from a lighthouse, so AB=AC.
Hence, ΔABC is an isosceles triangle.
The angle formed between the two boats, with the lighthouse as the vertex, measures 40°. So m∠A=40°.
The altitude to the base of an isosceles triangle bisects the vertex angle.
Hence, [tex]m\angle BAE=m\angle CAE=20^{\circ}[/tex]
The altitude to the base of an isosceles triangle bisects the base.
Hence, [tex]BE=CE=15[/tex]
In ΔABE,
[tex]\sin 20=\dfrac{BE}{AB}=\dfrac{15}{AB}[/tex]
[tex]\Rightarrow AB=\dfrac{15}{\sin 20}=43.86[/tex]
As AB=AC, so AC=43.86
