Two runners, P and Q, are on a circular track with a length of 4500 meters. P is running clockwise at a speed of 5 m/s, while Q is running counterclockwise at a speed of 7 m/s. Calculate the difference in the distance(in meters) covered by them when they meet for the tenth time.

The correct option is 7500.

Step-by-Step Explanation:

1. Understand the given data:
- Length of the circular track, $L=4500\text{meters}$
- Speed of runner P (running clockwise), $v_P=5\text{m/s}$
- Speed of runner Q (running counterclockwise), $v_Q=7\text{m/s}$

2. Relative speed of meeting:
Since P and Q are running in opposite directions (one clockwise, one counterclockwise), their relative speed is the sum of their individual speeds:

$v_{\text{rel}}=v_P+v_Q=5+7=12\text{m/s}$

3. Distance covered for meetings:
Every time they meet, the combined total distance covered by P and Q equals the full length of the track ($4500\text{meters}$).
Therefore, when they meet for the 10th time, the total combined distance covered by both runners is:

$\text{Total Distance}=10\times 4500=45000\text{meters}$

4. Time taken for the 10th meeting:
The time taken to cover this combined distance at the relative speed is:

$t=\frac{\text{Total Distance}}{v_{\text{rel}}}=\frac{45000}{12}=3750\text{seconds}$

5. Individual distances covered in 3750 seconds:
- Distance covered by P:

$D_P=v_P\times t=5\times 3750=18750\text{meters}$

- Distance covered by Q:

$D_Q=v_Q\times t=7\times 3750=26250\text{meters}$

6. Calculate the difference in distance covered:

$\text{Difference}=D_Q-D_P=26250-18750=7500\text{meters}$

Thus, the difference in the distance covered by P and Q when they meet for the tenth time is 7500 meters.