The first distance traveled is 15 mph*2 hours = 30 miles.
Let e = the horizontal distance between the starting point and the first turning point.
Let c = the vertical distance between the starting point and the first turning point.
There's a triangle defined by the sides e, c, and first distance traveled. Since it sailed at 30° and e and c are perpendicular, this is a 30°-60°-90° triangle.
A 30°-60°-90° triangle's sides has ratios of 1, √3, and 2, so we know e = 15√3 and c = 15.

Once the boat turns another 30°, the boat is at 60°. The second distance traveled is 4mph*2hours = 8 miles.
Let f = the horizontal distance between the first turning point and second turning point.
Let b = the vertical distance between the first turning point and the second turning point.
Again, there's a 30°-60°-90° triangle, so f = 4 and b = 4√3.

Now, the combined horizontal distance traveled, e+f, is 15√3+4 = 30.

After the boat turns another 120°, the boat is at 180°, going back towards the start horizontally. The last distance traveled is 15mph*2 hours = 30 miles. Thus, the boat has covered the full horizontal distance, e+f, by the end. Now we only need to find the vertical distance, c+b, to find how far the boat is from the start.

c+b = 15 + 4√3 = 21.93 miles