$\"\"$

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Consider the light house position is $\"\"$ .

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The light from a lighthouse can be seen from an $\"\"$ radius, i.e $\"\"$ .

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The two boats are $\"\"$ and $\"\"$ .

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Draw the diagram with above specifications.

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$\"\"$

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Consider triangle $\"\"$ .

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Law of sines : $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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The sum of the angles in any triangle is $\"\"$ .

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The two angles are $\"\"$ and $\"\"$ .

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The remaining angle $\"\"$ .

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$\"\"$ .

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$\"\"$

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Consider triangle $\"\"$ when the second boat enters into the radius of light house.

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Since triangle $\"\"$ is an isosceles triangle, therefore $\"\"$ .

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And $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Therefore the distance between the two boats when the second boat enters into the radius of light house is $\"\"$ i.e $\"\"$ .

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$\"\"$

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The distance between the two boats is $\"\"$ .