A state patrol officer saw a car start from rest at a highway on-ramp. She radio
ID: 2891539 • Letter: A
Question
A state patrol officer saw a car start from rest at a highway on-ramp. She radioed ahead to another officer 29 mi along the highway. When the car reached the location of the second officer 22 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 lm Why can the officer conclude that the driver exceeded the speed limit? Select the correct answer below and fill in the answer box to complete your choice. (Type an integer or decimal rounded to the nearest hundredth as needed.) O A. The officer can conclude the driver exceeded the speed limit because, by the Extreme Value Theorem, at some time c, the car was travelling at a speed of O B. The officer can conclude the driver exceeded the speed limit because, by Rolle's Theorem, at some time c, the car was travelling at a speed ofmiles per ° 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 miles per hour. hour. miles per hour.Explanation / Answer
Solution:
Speed = distance /time
speed = 29 miles/22 minute
speed = 29 miles /22 minute x 60 minutes/1 hour
speed = 870 miles/11 hours
speed = 79 mph
The speed actually exceed the limit.
I assume the speed limit is 60 mph. You can see that the person has travelled 29 miles in 22 minutes,
this is an average (mean) speed of ~ 62 mph.
The mean value theorem simply says that at some point between the two endpoints, the car must have been traveling at this average speed.
The function in question is x vs t where t is time and x is the distance from the on ramp.
You know
x(0) = 0 and x(22 min) = 29 miles.
Calculate the slope of a line between the two endpoints and then apply the theorem. It doesn't matter what the actual curve (as long as cont, diff, etc)
On the real line
Rolles theorem says if your car is in the same place today as it was yesterday, it must have been stopped sometime in between.
The extreme value theorem says that on a closed interval of time there must be a maximum distance the car was from your house.
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