Answer
$799.9\ mi$.
Work Step by Step
Step 1. Draw a diagram as shown in the figure. Based on the given bearings and distances, we have $AB=580, BC=400\ mi$.
Step 2. In triangle ABC, we have the angle
$B=34^\circ+74^\circ=108^\circ$
Step 3. Using the Law of Cosines, let $AC=b$. We have
$b^2=580^2+400^2-2(580)(400)cos(108^\circ)\approx639,784$
which gives $c\approx799.9\ mi$.
