Mean Value Theorem and the police A state patrol officer saw a...

Question

Mean Value Theorem and the police A state patrol officer saw a car start from rest at a highway on-ramp. She radioed ahead to a patrol officer 30 mi along the highway. When the car reached the location of the second officer 28 min later, it was clocked going 60 mi/hr. The driver of the car was given a ticket for exceeding the 60-mi/hr speed limit. Why can the officer conclude that the driver exceeded the speed limit?

Accepted Answer

Welcome back, everyone. In this problem, a cyclist starts from rest at the beginning of a 50 kilometer bike trail. An observer at the end of the trail notes that the cyclist arrives 1.5 hours later, traveling at a speed of 25 kilometers per hour. If the speed limit on the trail is 25 kilometers per hour, how can the observer conclude that the cyclist exceeded the speed limit? A says the cyclist was observed traveling at 25 kilometers per hour. B The cyclist traveled 50 kilometers in 1.5 hours. The mean value theorem indicates the cyclist's speed exceeded 25 kilometers per hour at some point. And the D says the cyclist's average speed was 25 kilometers per hour, so the cyclist did not exceed the speed limit. No If we're gonna figure out how we can show that the cyclists exceeded the speed limit, let's first make note of all the information we have. What do we already know? Well, so far we know that the distance of the bike trail, let's call that D is equal to 50 kilometers. We also know that the time taken by the cyclist T is 1.5 hours. The average speed of the cyclist, OK, it's called the average is 25 kilometers per hour. And the speed limit on the trail, let's call that VIM, is 25 kilometers per hour. OK? Now, even though the cyclist was observed traveling at a speed of 25 kilometers per hour, let's see if we can figure out the cyclist's average speed. Let's take into consideration what would have happened while he was riding on the bike trail. Now we can use the formula or the equation for average speed. Recall that average speed would be equal to the change in distance divided by the change in time. This change in distance is 50 minus 0, and the change in time is 1.5 minus 0 because it took 1.5 hours. So you could just write here 1.5. 50 divided by 1.5 equals 1003, OK? Or 333 kilometers per hour. No, since the average speed of the cyclist is calculated to be 1003 kilometers per hour, it's greater than the speed limit of 25 kilometers per hour. Thus, the observer that can conclude that the cyclist must have exceeded the speed limit at some point during the trip. Therefore, C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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