Let BE=X, because it takes 20 minutes from B to E and 80 minutes from C to B, so CB=4X.

Another angle BAE = 30, cosine theorem can get an equation between AB and X, another angle BAC =120, cosine theorem can get an equation between AC and X, let the extension line of CD perpendicular to EA be at point D, and the angle CAD = 30, sine theorem can get the equation about X between CD and AD.

Now EC, CD and ED can all be expressed by formulas containing X, and triangle ECD is a right triangle, so Pythagorean theorem can get X.

I can't draw it, but it should be the same as yours. I hope it helps you.