Final answer:
Using the Mean Value Theorem, the car's acceleration is found to be approximately 0.0037253 m/s², assuming a constant increase in speed from 30mph to 35mph over a 10-minute period.
Step-by-step explanation:
The question is asking to calculate the acceleration of a car given the change in speed over a certain time interval, using the Mean Value Theorem (MVT) from calculus.
According to the Mean Value Theorem, there exists some point t in the interval where the instantaneous rate of change (acceleration in this case) is the same as the average rate of change over the interval. The car's speed changes from 30 mph to 35 mph over 10 minutes (1/6 hour). Therefore, the average acceleration a can be calculated using the formula:
a = (change in velocity) / (change in time)
First, we need to convert the change in velocity to consistent units:
5 mph * (1609.34 meters/1 mile) * (1 hour/3600 seconds) = 2.2352 meters/second
Now, we calculate the acceleration:
a = 2.2352 m/s / (10 minutes * 60 seconds/minute)
a = 0.0037253 m/s²
So, the car must have achieved an acceleration of 0.0037253 m/s² at some point within the 10-minute interval to go from 30mph to 35mph.
Note: The acceleration calculated is a rough estimate and assumes a fairly constant rate of increase, as required by the Mean Value Theorem.