Faced with a trip of 100 miles and the need to arrive at my destination by 3:30pm, I deliberately chose to leave home at 1:30. Experience has taught me that while driving, I can comfortably average 50 miles per hour from start to finish. In answer to Question 1, the following calculation revealed the time needed to complete the journey.
Your average driving speed may be faster than mine. (Don't forget to include pit stops!) Type in your average speed to check how long it would take for you to drive from Corvallis to Portland.
For example, you should find that at an average driving speed of 60 miles per hour, one could spend up to 20 minutes with a police officer and still complete the trip in two hours.
These calculations involve three quantities: elapsed time, distance traveled, and average speed. The following formula captures the relationship between these variables.
Distance traveled is a function of elapsed time. Measuring these quantities with respect to my instant of departure at 1:30pm on that November afternoon, we set
Average speed can be expressed in terms of the distance function. Consider a time interval that begins at time t0 and ends at time t1. For such a time interval, the definition of average speed can be expressed in terms of the distance function as follows.
A difference quotient for the distance function determines average speed for a suitable time interval. Average speed is the key to Question 1. When it comes to Question 2, a closer look at the distance function will reveal how the police officer was able to clock my speed at 38 miles per hour. So get out your stopwatch and let's consider how the police use radar technology to catch speeders!
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William A. Bogley