How do I explain this problem?
A trucker handed in a ticket at a toll booth showing that in 2 h he had covered 159 mi on a toll road with speed limit 65 mph. The trucked was cited for speeding, why?
I know it is mean value theorem, but how?
His average speed was 159/2 = 79mph. There is no way he can have an average greater than the max speed limit w/o speeding.
To explain why the trucker was cited for speeding, we can use the Mean Value Theorem (MVT). The MVT is a principle in calculus that states that if a function (in this case, the truck's position as a function of time) is continuous on a closed interval, and differentiable on the open interval, then there exists at least one point within that interval where the instantaneous rate of change (velocity) equals the average rate of change.
In this problem, the trucker traveled a distance of 159 miles in 2 hours on a toll road with a speed limit of 65 mph. To find the trucker's average speed, we divide the total distance traveled (159 miles) by the total time taken (2 hours), giving us an average speed of 79 mph.
According to the MVT, since the trucker's average speed exceeds the speed limit of 65 mph, there must have been a point during the 2-hour interval where the trucker's instantaneous speed was equal to or greater than 79 mph. This implies that the trucker exceeded the speed limit at some point during the journey and hence was cited for speeding.
Therefore, using the Mean Value Theorem, we can conclude that the trucker was cited for speeding because the average speed exceeded the speed limit, indicating a violation of the law.