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The problem describes the motion of a swimmer who swims for 10 minutes. In the first 6 minutes, he swims 80 meters per minute, and after the sixth minute, he swims half the distance per minute compared to the previous one. We need to:

1. Define the function for the total distance \( S(t) \) over time \( t \), where \( 0 \leq t \leq 10 \).
2. Calculate the distance the swimmer covers in 9 minutes.

### Step 1: Define the function \( S(t) \)

- **For the first 6 minutes (0 ≤ t ≤ 6):** The swimmer covers 80 meters per minute. Therefore, the distance covered in this period is:

  \[
  S(t) = 80t \quad \text{for} \quad 0 \leq t \leq 6
  \]

- **For the last 4 minutes (6 < t ≤ 10):** After the sixth minute, the swimmer covers half the distance per minute of the previous minute's distance. This creates a geometric progression. The distances swum in minutes 7, 8, 9, and 10 are \( 40 \, m \), \( 20 \, m \), \( 10 \, m \), and \( 5 \, m \), respectively.

  The sum of the distances swum after minute 6 is:

  \[
  \text{Sum of distances from 7 to 10} = 40 + 20 + 10 + 5 = 75 \, \text{m}
  \]

  The total distance covered in 10 minutes is:

  \[
  S(t) = 480 + 75 = 555 \, \text{m}
  \]

### Step 2: Calculate the distance swum in 9 minutes

- For the first 6 minutes, the swimmer covers:

  \[
  80 \times 6 = 480 \, \text{m}
  \]

- In minute 7, he swims 40 meters.
- In minute 8, he swims 20 meters.
- In minute 9, he swims 10 meters.

The total distance covered in 9 minutes is:

\[
S(9) = 480 + 40 + 20 + 10 = 550 \, \text{m}
\]

### Final Answer:
The swimmer covers **550 meters** in 9 minutes.

Would you like a further breakdown of the geometric progression or have any questions?

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Here are 5 related questions:
1. How would the problem change if the swimmer slowed down by one-third instead of half after the sixth minute?
2. How can we generalize this function if the initial speed is different?
3. What would the total distance be if the swimming session lasted 12 minutes?
4. How would the function change if the swimmer sped up after the sixth minute instead of slowing down?
5. Can you derive a formula for the total distance swum in the first 6 minutes for any given speed?

**Tip:** Geometric series are often used when distances or speeds decrease at a constant ratio after a certain point!

A swimmer swims for 10 minutes. In the first six minutes, he swims 80 meters per minute. After the sixth minute, he swims at half the rate compared to the previous minute. Define a function for the total distance S(t), where 0 ≤ t ≤ 10. Calculate the distance the swimmer has covered by the 9th minute.

This problem involves calculating the total distance swum by a swimmer in 9 minutes using geometric progression. The swimmer starts at a constant rate of 80 meters per minute and then slows down by half after 6 minutes. Suitable for grades 9-11 students learning piecewise functions and geometric series.

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