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Car B starts at point X and moves clockwise around a circula

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Car B starts at point X and moves clockwise around a circula

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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)?

A. 4pi - 1.6
B. 4pi + 8.4
C. 4pi + 10.4
D. 2pi - 1.6
E. 2pi - 0.8

OA B

Source: Manhattan Prep

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BTGmoderatorDC wrote:
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)?

A. 4pi - 1.6
B. 4pi + 8.4
C. 4pi + 10.4
D. 2pi - 1.6
E. 2pi - 0.8
π ≈ 3.

Car B is traveling for more than 10 hours, so answer choices D and E are too small, and A is unlikely. The correct answer is either B or C.

Circumference of track = 20π ≈ 60 miles.
In 10 hours, distance for B = 2*10 = 20 miles.
60-20 = 40 miles between A and B.
A and B now have to travel the 40 miles between them and then -- after they meet -- keep traveling in opposite directions until there is another 12 miles between them. So the total distance that they need to travel is 40+12 = 52 miles.
When things work together, we can add their rates. Combined rate for A+B = 3+2 = 5 miles/hour.
Time for A+B = Distance/Rate = 52/5 = 10.4 hours.
Since B traveled for 10 hours before A joined in, the total time for B = 10 + 10.4 = 20.4 hours.
Only B works: 4π + 8.4 ≈ 12 + 8.4 = 20.4.

The correct answer is B.

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