Answer:
603.26 miles.
Step-by-step explanation:
By the cosine law, we can find the distance from the port to the island as follows

Where c is the distance from the port to the island, a and b are the other sides of the triangle with lengths 400 mi and 250 mi and C is the angle between them of 135 degrees.
So, replacing the values, we get:
![\begin{gathered} c^2=400^2+250^2-2(400)(250)\cos 135_{} \\ c^2=160000+62500-(200000)(-0.7071) \\ c^2=160000+62500+141421.35 \\ c^2=363921.35 \\ c=\sqrt[]{363921.35} \\ c=603.26\text{ mi} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/m15x58gg49l47t9atoa3y1v47z4weingwf.png)
Therefore, the approximate distance between the port and the island is 603.26 miles.