At 2:00 pm a car's speedomoter reads 20 mph, and at 2:10 pm it reads 25 mph. Use the Mean Value Theorem to find an acceleration the car must achieve.
at some time between 2:00 and 2:10 if it is continuous and differentiable the acceleration must hit the average acceleration.
(25 -20) mph/(10/60 hour)
or in more usual units
(36.67 - 29.33) ft/sec/600 seconds
To find the acceleration using the Mean Value Theorem, we need to find the average rate of change of the car's speed over the given time interval.
The Mean Value Theorem states that if a function is continuous over a closed interval and differentiable over an open interval, then there exists at least one point in the open interval where the instantaneous rate of change is equal to the average rate of change over the closed interval.
Let's denote the initial time as t1 = 2:00 pm and the final time as t2 = 2:10 pm.
The initial speed at t1 is v1 = 20 mph, and the final speed at t2 is v2 = 25 mph.
The average rate of change (acceleration) over the interval [t1, t2] is given by the formula:
acceleration = (v2 - v1) / (t2 - t1)
Substituting the given values:
acceleration = (25 mph - 20 mph) / (2:10 pm - 2:00 pm)
To compute the time difference (t2 - t1), convert the time interval 2:10 pm - 2:00 pm to minutes:
10 minutes - 0 minutes = 10 minutes
Now, substitute the values into the formula:
acceleration = (25 mph - 20 mph) / 10 minutes
Simplify the expression:
acceleration = 5 mph / 10 minutes
The units of acceleration are miles per hour per minute. To simplify further, we can convert minutes to hours:
1 hour = 60 minutes
Therefore, 10 minutes is equal to 10/60 hours.
acceleration = 5 mph / (10/60) hours
acceleration = 30 mph / 10 hours
Finally, we have the acceleration:
acceleration = 3 mph/h
So, the car must achieve an acceleration of 3 mph/h.