An airplane is flying at an altitude of 10,000 metres. From the cockpit, the pilot spots an airport tower, which is located at point A on the ground. The pilot notes that the angle of depression from the airplane to the tower is 30 degrees. The airplane then turns to a bearing of 120 degrees and flies for 15 kilometres, reaching point $B$.

From point B, the pilot spots a runway, which is located at point $C$ on the ground. The angle of depression from the airplane to the runway is 45 degrees. The pilot decides to land the plane at the runway.
Calculate the distance, in miles, between the tower at point $A$ and the runway at point $C$.

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 ：\(\boxed{43495.97}\)