Two autonomous rovers, Alpha and Beta, are dispatched from the same starting point on a circular circuit measuring 2400 meters in length. Rover Alpha travels clockwise at a constant speed of 6 m/s, whereas Rover Beta travels counterclockwise at a constant speed of 10 m/s. What is the difference in the total distance (in meters) covered by the two rovers when they cross paths for the eighth time?

The correct answer is 4800.

Step 1: Understand the motion of both rovers
Rover Alpha travels clockwise around the circular circuit at a constant speed of 6 m/s, while Rover Beta travels counterclockwise at a constant speed of 10 m/s. Since they start from the same point and travel in opposite directions along the circular path, they move toward each other, reducing the distance between them.

Step 2: Calculate the relative speed
Because the rovers move in opposite directions, their relative speed of approach is the sum of their individual speeds:

$v_{\text{relative}}=v_{\text{Alpha}}+v_{\text{Beta}}=6\text{m/s}+10\text{m/s}=16\text{m/s}$

Step 3: Determine the total combined distance for 8 crossings
Each time the two rovers cross paths, the sum of the distances covered by both rovers equals one complete round of the circuit (2400 meters).
Therefore, when they cross paths for the 8^th time, their total combined distance traveled is:

$\text{Total Combined Distance}=8\times 2400\text{m}=19200\text{m}$

Step 4: Calculate the total time elapsed
The time $t$ required to complete 8 crossings is the total combined distance divided by their relative speed:

$t=\frac{19200\text{m}}{16\text{m/s}}=1200\text{seconds}$

Step 5: Calculate individual distances traveled
Distance covered by Rover Alpha ($d_{\text{Alpha}}$):

$d_{\text{Alpha}}=6\text{m/s}\times 1200\text{s}=7200\text{meters}$

Distance covered by Rover Beta ($d_{\text{Beta}}$):

$d_{\text{Beta}}=10\text{m/s}\times 1200\text{s}=12000\text{meters}$

Step 6: Find the difference in distance covered
Subtract the distance covered by Rover Alpha from the distance covered by Rover Beta:

$\text{Difference}=d_{\text{Beta}}-d_{\text{Alpha}}=12000\text{m}-7200\text{m}=4800\text{meters}$

Thus, the difference in total distance covered by the two rovers when they cross paths for the eighth time is 4800 meters.