To analyze this problem, let’s break it down step-by-step:

Given Information

1. A car started from rest at a highway on-ramp.
2. The car traveled 21 miles along the highway.
3. The time taken to cover this distance was 19 minutes.
4. At the end of this distance, the car was clocked at 60 mph.
5. The speed limit is 60 mph, and we need to determine if the car exceeded this limit at any point.

Key Concepts and Calculation

This problem is about proving that the driver exceeded the speed limit using the Mean Value Theorem (MVT) from calculus.

Step-by-Step Solution

1. Mean Value Theorem Setup

The Mean Value Theorem states that if a function $f(x)$ is continuous on a closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, then there exists some point $c$ in $(a, b)$ such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$

In this case, $f(t)$ represents the position of the car at time $t$, and $f'(t)$ (the derivative of position) represents the car's speed.

2. Applying the Mean Value Theorem

Since the car traveled 21 miles in 19 minutes, we calculate the average speed over this interval: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$

Convert 19 minutes into hours: $19 \text{ minutes} = \frac{19}{60} \text{ hours}$

Now, calculate the average speed: $\text{Average Speed} = \frac{21 \text{ miles}}{\frac{19}{60} \text{ hours}} = 21 \times \frac{60}{19} \approx 66.32 \text{ mph}$

3. Interpretation

The Mean Value Theorem implies that at some point during the journey, the car's speed must have been equal to this average speed of 66.32 mph, which exceeds the speed limit of 60 mph.

Conclusion

Thus, the correct answer is: C. "The officer can conclude the driver exceeded the speed limit because, by the Mean Value Theorem, at some time $c$, the car was travelling at a speed of 66.32 miles per hour."

Would you like additional details or have any questions?

Further Questions

1. What conditions are required to apply the Mean Value Theorem?
2. How would the answer change if the average speed was calculated to be exactly 60 mph?
3. What if the car was observed at various speeds during the journey? How would this affect the conclusion?
4. Can the Mean Value Theorem be applied if the car’s speed was variable throughout the journey?
5. How is the Mean Value Theorem different from the Extreme Value Theorem in this context?

Tip

Remember, the Mean Value Theorem helps identify points where instantaneous rates (like speed) match the average over a given interval.