Answer
The ship is 2.4 miles from the lighthouse.
Work Step by Step
Let A be the ship's first position.
Let B be the ship's final position.
Let C be the position of the lighthouse.
The points ABC form a triangle.
The angle $A = 30^{\circ}$
The angle $B = 180^{\circ}-55^{\circ} = 125^{\circ}$
The angle $C = 180^{\circ}-30^{\circ}-125^{\circ} = 25^{\circ}$
Let $c = 2.0~mi$. Let $a$ be the distance from the ship's final position to the lighthouse. We can use the law of sines to find $a$:
$\frac{a}{sin~A} = \frac{c}{sin~C}$
$a = \frac{c~sin~A}{sin~C}$
$a = \frac{(2.0~mi)~sin~30^{\circ}}{sin~25^{\circ}}$
$a = 2.4~mi$
The ship is 2.4 miles from the lighthouse.
