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Let's break down the problem step by step.

### Given Information:
- The race takes place on a 400-meter track.
- Two athletes are running at constant, but different speeds.
- The faster athlete completes each lap in **66 seconds**.
- After running **17.5 laps**, the faster athlete overtakes the slower athlete for the first time.

### a) Find the time the slower athlete takes to complete each lap.

We know that the faster athlete overtakes the slower one after 17.5 laps. This means the slower athlete has completed 16.5 laps in the same amount of time (since they meet for the first time after this difference of 1 lap).

Let the time the slower athlete takes to complete one lap be \( t_s \) (in seconds).

- The time the faster athlete takes for 17.5 laps is:
  \[
  t_f = 17.5 \times 66 = 1155 \text{ seconds}
  \]
- In this same time, the slower athlete completes 16.5 laps:
  \[
  16.5 \times t_s = 1155 \text{ seconds}
  \]
Thus, the time per lap for the slower athlete is:
\[
t_s = \frac{1155}{16.5} = 70 \text{ seconds}
\]

### b) Time for the faster athlete to complete the 10,000 meters and the distance covered by the slower athlete.

- **Time for 10,000 meters**: Since each lap is 400 meters, the number of laps in 10,000 meters is:
  \[
  \frac{10,000}{400} = 25 \text{ laps}
  \]
  The time taken by the faster athlete to complete 25 laps is:
  \[
  t = 25 \times 66 = 1650 \text{ seconds}
  \]

- **Distance covered by the slower athlete** in this time: The slower athlete's speed is one lap every 70 seconds. So, in 1650 seconds, the slower athlete covers:
  \[
  \frac{1650}{70} = 23.57 \text{ laps}
  \]
  The distance covered by the slower athlete is:
  \[
  23.57 \times 400 = 9428 \text{ meters}
  \]

### Summary:
- a) The slower athlete takes **70 seconds** per lap.
- b) The faster athlete completes the 10,000 meters in **1650 seconds**. By that time, the slower athlete has covered **9428 meters**.

Would you like a more detailed explanation or have any further questions? Here are some related questions:

1. How would the problem change if the track were longer or shorter than 400 meters?
2. What if the faster athlete had overtaken the slower athlete after 20 laps?
3. How would this problem be solved if the overtaking happened after a different number of laps?
4. What is the average speed of each athlete in meters per second?
5. How long would it take for the slower athlete to complete the 10,000 meters?

**Tip:** To find when one object overtakes another, compare the difference in their distances over time.

Two athletes started side by side in a race on a 400-meter track. The faster athlete completed each lap in 66 seconds. After 17.5 laps, the faster athlete overtook the slower athlete for the first time. Based on this, how long did it take the slower athlete to complete each lap and how much time did it take the faster athlete to complete 10,000 meters?

This problem involves calculating the time taken by two athletes running at different speeds on a 400-meter track. It explores the concept of laps, speed, and time, and provides a detailed step-by-step solution to determine the slower athlete's lap time and the distance covered by the slower athlete when the faster athlete completes 10,000 meters.

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