Step 1 :\(\tan 30^\circ = \frac{10000}{AC}\)
Step 2 :\(AC = \frac{10000}{\tan 30^\circ} = 10000 \times \sqrt{3}\)
Step 3 :\(\tan 45^\circ = \frac{10000}{BC}\)
Step 4 :\(BC = \frac{10000}{\tan 45^\circ} = 10000\)
Step 5 :Use the Law of Cosines on \(\triangle ABC\)
Step 6 :\(AB^2 = AC^2 + BC^2 - 2 \times AC \times BC \times \cos 120^\circ\)
Step 7 :\(AB^2 = (10000 \times \sqrt{3})^2 + 10000^2 - 2 \times (10000 \times \sqrt{3}) \times 10000 \times (-\frac{1}{2})\)
Step 8 :\(AB^2 = 30000^2 + 10000^2 + 30000^2\)
Step 9 :\(AB^2 = 70000^2\)
Step 10 :\(AB = 70000\)
Step 11 :Convert kilometers to miles: \(1 \text{ km} = 0.621371 \text{ miles}\)
Step 12 :\(AB = 70000 \times 0.621371 = 43495.97\)
Step 13 :\(\boxed{43495.97}\)