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This problem involves the Mean Value Theorem (MVT) and requires proving that the sports car exceeded the speed limit at some point between two radar points. Here's a breakdown of the solution:

### Given:
- Two radar patrol cars are 6 miles apart.
- The speed limit is 65 mph.
- The sports car passes the first radar going 60 mph.
- It passes the second radar 5 minutes later going 65 mph.

### Mean Value Theorem (MVT):
The Mean Value Theorem states that for a function \( f(x) \) that is continuous on the closed interval \( [a, b] \) and differentiable on the open interval \( (a, b) \), there is at least one point \( c \in (a, b) \) such that:

\[
f'(c) = \frac{f(b) - f(a)}{b - a}
\]

In the context of this problem:
- The function \( f(t) \) represents the position of the car over time.
- The derivative \( f'(t) \) represents the velocity (speed) of the car.
- We need to analyze the car's average speed between the two radar points and show that at some point, it must have been speeding.

### Solution Steps:

1. **Convert time to hours:**
   The car takes 5 minutes between the two radar points. Since speed is in miles per hour (mph), convert 5 minutes into hours:
   \[
   \text{Time} = \frac{5}{60} = \frac{1}{12} \, \text{hours}
   \]

2. **Calculate average speed:**
   The car traveled 6 miles in \( \frac{1}{12} \) hours. The average speed between the two radar points is:
   \[
   \text{Average speed} = \frac{6 \, \text{miles}}{\frac{1}{12} \, \text{hours}} = 6 \times 12 = 72 \, \text{mph}
   \]

3. **Application of MVT:**
   Since the average speed between the two points is 72 mph, by the Mean Value Theorem, there must be at least one point between the two radar clocks where the car's instantaneous speed was exactly 72 mph. Since 72 mph exceeds the speed limit of 65 mph, this shows that the car must have been speeding at some point.

Thus, the sports car exceeded the speed limit between the two clockings.

Would you like more details on any step?

### Five related questions:
1. How does the Mean Value Theorem apply in this scenario?
2. Why is the average speed important in proving the car was speeding?
3. What assumptions do we make about the car's speed between the two radar points?
4. How can we calculate instantaneous velocity from position using calculus?
5. What other real-life scenarios could we apply the Mean Value Theorem to?

### Tip:
When solving problems with the Mean Value Theorem, always check that the function meets the criteria: it must be continuous and differentiable over the relevant interval.

Two radar patrol cars are located at fixed positions 6 miles apart on a long, straight road where the speed limit is 65 miles per hour. A sports car passes the first patrol car traveling at 60 miles per hour, then 5 minutes later, it passes the second patrol car going 65 miles per hour. Analyze this situation to show that at some time between the two clockings, the sports car exceeded the speed limit. Hint: Use the MVT.

Explore how the Mean Value Theorem applies to a speeding problem involving two radar points. The problem analyzes a sports car's travel over 6 miles, demonstrating that at some point the car exceeded the 65 mph speed limit. This solution involves calculus concepts, such as velocity and average speed, and is suitable for high school or college-level students.

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