Solution to warm-up 2:
In the frame of reference of the disk, the shortest distances are straight lines. That is, the shortest path from one point on a circle to another is a straight-line segment. In general, the car will therefore traverse a chord through the disk such that the end of the chord is precisely the point where the car wants to exit given the disk's rotation. See this illustration:
For example, if the car needs to go 130 degrees around the disk in the nonrotational frame, then it might be possible for the car to traverse a chord that subtends, say, 80 degrees at the same time the disk rotates 50 degrees.
That is, the fastest way to get from one point to the other in the disk is to aim the chord at a closer point and then let the disk swing the car around to the proper point.
End of Warm-up 2