Answer
The initial distance between the ship and the lighthouse is 5.1 miles
The final distance between the ship and the lighthouse is 7.2 miles
Work Step by Step
Let $A$ be the first position of the ship. Then angle $A = 180^{\circ} - 37^{\circ}$ which is $143^{\circ}$.
Let $C$ be the final position of the ship. Then angle $C = 25^{\circ}$.
Let the lighthouse be located at the position of angle $B$. We can find angle $B$:
$A+B+C = 180^{\circ}$
$B = 180^{\circ}-A-C$
$B = 180^{\circ}-143^{\circ}-25^{\circ}$
$B = 12^{\circ}$
We can find the length of side $c$ which is the initial distance between the ship and the lighthouse:
$\frac{c}{sin~C} = \frac{b}{sin~B}$
$c = \frac{b~sin~C}{sin~B}$
$c = \frac{(2.5~mi)~sin~(25^{\circ})}{sin~(12^{\circ})}$
$c = 5.1~mi$
The initial distance between the ship and the lighthouse is 5.1 miles
We can find the length of side $a$ which is the final distance between the ship and the lighthouse:
$\frac{a}{sin~A} = \frac{b}{sin~B}$
$a = \frac{b~sin~A}{sin~B}$
$a = \frac{(2.5~mi)~sin~(143^{\circ})}{sin~(12^{\circ})}$
$a = 7.2~mi$
The final distance between the ship and the lighthouse is 7.2 miles
