Answer:
<em>Each boat are </em><em>43.86</em><em> miles from the lighthouse.</em>
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, ![m\angle BAE=m\angle CAE=20^{\circ}](https://tex.z-dn.net/?f=m%5Cangle%20BAE%3Dm%5Cangle%20CAE%3D20%5E%7B%5Ccirc%7D)
The altitude to the base of an isosceles triangle bisects the base.
Hence, ![BE=CE=15](https://tex.z-dn.net/?f=BE%3DCE%3D15)
In ΔABE,
![\sin 20=\dfrac{BE}{AB}=\dfrac{15}{AB}](https://tex.z-dn.net/?f=%5Csin%2020%3D%5Cdfrac%7BBE%7D%7BAB%7D%3D%5Cdfrac%7B15%7D%7BAB%7D)
![\Rightarrow AB=\dfrac{15}{\sin 20}=43.86](https://tex.z-dn.net/?f=%5CRightarrow%20AB%3D%5Cdfrac%7B15%7D%7B%5Csin%2020%7D%3D43.86)
As AB=AC, so AC=43.86