Final answer:
The distance between the two ships is approximately 390.51 km apart. This is calculated using the Pythagorean theorem, as their paths form a right triangle with legs of 300 km northwest and 250 km east.
Step-by-step explanation:
To determine how far apart two ships are after one sails 300 km northwest and the other 250 km east, we can use the Pythagorean theorem. This is because the paths they travel form a right triangle, with one ship's displacement being one leg of the triangle, and the other ship's displacement being the other leg.
First, let's define the directions as vectors. The ship that sails 300 km northwest can be represented as a vector pointing 45° west of north because northwest is exactly between north and west. On the other hand, the ship sailing 250 km east is represented by a vector pointing directly along the east axis.
Since these directions are perpendicular, we can find the distance between the two ships by calculating the hypotenuse of the right triangle formed by their paths using the Pythagorean theorem:
c = √(a² + b²)
where 'a' is the distance one ship traveled northwest (300 km), and 'b' is the distance the other ship traveled east (250 km). Now we can calculate:
c = √(300² + 250²)
c = √(90000 + 62500)
c = √(152500)
c = 390.51 km (rounded to two decimal places)
So, the two ships are approximately 390.51 km apart from each other.