To solve the problem, let's break down the situation step by step:

1. First 10 minutes:
   John travels at a rate of 1 mile per minute. So, in the first 10 minutes, he covers: $1 \text{ mile/minute} \times 10 \text{ minutes} = 10 \text{ miles}$ After 10 minutes, John has traveled 10 miles.

2. 15-minute stop:
   John stops at a restaurant for 15 minutes. During this time, he does not cover any distance, so his total distance remains at 10 miles.

3. Next 10 minutes:
   After the stop, John resumes driving but at a slower pace of 0.5 miles per minute. In the next 10 minutes, he covers: $0.5 \text{ mile/minute} \times 10 \text{ minutes} = 5 \text{ miles}$ After these 10 minutes, John has covered a total of: $10 \text{ miles} + 5 \text{ miles} = 15 \text{ miles}$

Now let's analyze each option:

• Option A: The graph rises to 5 miles in the first 10 minutes, stays at 5 miles for the next 15 minutes, and then rises to 15 miles by the 35th minute. This does not match our situation since John covers 10 miles in the first 10 minutes, not 5 miles.

• Option B: The graph rises to 10 miles in the first 10 minutes, stays at 10 miles during the stop, but then declines to 5 miles. This is incorrect since John's distance should increase, not decrease.

• Option C: The graph rises to 10 miles in the first 10 minutes, stays at 10 miles during the stop, and then rises to 15 miles by the 35th minute. This accurately represents John's travel, where he covers 10 miles initially, pauses, and then covers an additional 5 miles.

• Option D: The graph rises to 10 miles in the first 10 minutes, stays at 10 miles during the stop, and then rises to 20 miles by the 35th minute. This suggests John covers 10 miles in the last 10 minutes, which is incorrect since he only covers 5 miles.

Correct Answer: C
The graph in Option C accurately models the piecewise function for John's trip.

Would you like more details, or have any questions?

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Here are 5 questions related to this problem:

1. How would the graph change if John stopped for 20 minutes instead of 15 minutes?
2. What would the graph look like if John accelerated to 2 miles per minute after his stop?
3. How would the graph differ if John had no stop and drove straight to his office?
4. If John covered 0.25 miles per minute after his stop, what would be the final distance covered by the 35th minute?
5. How does the graph represent the concept of a piecewise function in this scenario?

Tip: When analyzing a piecewise function, always break down the problem into distinct intervals and analyze each part separately.