To solve this problem, let's break down the information given and apply the Mean Value Theorem (MVT) as suggested by the question.

Given Information:

1. A car starts from rest at a highway on-ramp.
2. The car travels 25 miles along the highway, reaching a second officer after 21 minutes.
3. The speed limit is 60 miles per hour.
4. The officer gave a ticket for exceeding this speed limit.

Mean Value Theorem:

The Mean Value Theorem (MVT) in calculus states that if a function $f$ is continuous on a closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, then there exists at least one point $c$ in $(a, b)$ where: $f'(c) = \frac{f(b) - f(a)}{b - a}$ For this scenario, $f(b) - f(a)$ represents the total distance traveled (25 miles) and $b - a$ represents the time taken (21 minutes).

Converting Time:

Since the speed limit is in miles per hour, we need to convert 21 minutes into hours: $21 \text{ minutes} = \frac{21}{60} \text{ hours} = 0.35 \text{ hours}$

Applying MVT:

Using the MVT, the average speed over the interval is: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{25 \text{ miles}}{0.35 \text{ hours}} = 71.43 \text{ miles per hour}$

Conclusion:

Since 71.43 miles per hour is greater than the speed limit of 60 miles per hour, the Mean Value Theorem guarantees that at some point during the trip, the car must have been traveling at this speed (or higher), which exceeds the speed limit.

Answer:

The correct choice is:

• Option C: "The officer can conclude the driver exceeded the speed limit because, by the Mean Value Theorem, at some time $c$, the car was traveling at a speed of 71.43 miles per hour."

Would you like more details on the Mean Value Theorem or how this conclusion was derived?

Related Questions:

1. How does the Mean Value Theorem apply to real-life scenarios beyond speed limits?
2. What is the difference between the Mean Value Theorem and Rolle's Theorem?
3. How do you convert minutes into hours in speed calculations?
4. Why is it necessary to consider average speed rather than instantaneous speed?
5. In what situations would the Extreme Value Theorem be more applicable?

Tip:

When solving problems involving speed and distance, always ensure units are consistent, especially for time (e.g., convert minutes to hours if speed is in miles per hour).