Answer
The distance between the two ships is 745 miles.
Work Step by Step
Let $a = 402~mi$, let $b = 402~mi$, and let angle $C = 135^{\circ}40'$.
We can use the law of cosines to find $c$, the length of the line opposite the angle $C$:
$c^2 = a^2+b^2-2ab~cos~C$
$c = \sqrt{a^2+b^2-2ab~cos~C}$
$c = \sqrt{(402~mi)^2+(402~mi)^2-(2)(402~mi)(402~mi)~cos~135^{\circ}40'}$
$c = \sqrt{554395.6~mi^2}$
$c = 745~mi$
The distance between the two ships is 745 miles.
