Car b starts at point x and moves clockwise around a circular track at a constant rate of 2 mph. ten hours later, car a leaves from point x and travels counter-clockwise around the same circular track at a constant rate of 3 mph. if the radius of the track is 10 miles, for how many hours will car b have been traveling when the cars have passed each other for the first time and put another 12 miles between them (measured around the curve of the track)?

20.96 h

Explanation:

The perimeter of the track is 2*pi*r = 20pi miles

In 10 hours, car B would have moved 20miles. So, when Car A leaves from point X, car B is 20pi - 20 miles from point X counter-clockwise and car A.

From here, we can express the distance of A from X like this:

xa = 3t

And the distance of B would be:

xb = 20pi - 20 - 2t

The time t where they would passed each other and put  12 miles between them would be the one where xa - xb is equal to 12:

xa - xb = 12

3t - (20pi - 20 - 2t) = 12

5t = 20 pi - 8

t = (20pi - 8)/5 = 10.96 h

Remember to add this value to the 10 hours car B had already been racing:

t = 20.96h