Greenwood press

98
RATES
The relationship between the distance
an automobile travels compared to its
time when it travels at a constant rate
of 55 miles per hour.
A distance versus time graph of
an automobile with varying
rates during a three-hour time
period. The slope of the dotted
line is the automobile’s average
speed during that time period.

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speeding ticket without having been tracked by a radar speed-detection device! In
this case, the change in distance between tollbooths 3 and 17 is 109 miles, and the
change in time between 12:34 
PM
and 1:57 
PM
is 1 hour and 23 minutes, or approx-
imately 1.38 hours. Therefore the average speed of the car is about 79 miles per
hour, much faster than the speed limit! The mean-value theorem in calculus
implies that a car constantly in motion with this average speed will have traveled
at that rate at least one time during the journey, even if undetected by radar. The
graph below describes the position of the car for its time on the highway. The dot-
ted line represents the average rate of 79 miles per hour. The three times that the
car was traveling at 79 miles per hour are indicated with the word “speeding.”
Note that there are many other times that the car was speeding more than 79 miles
per hour. The mean value theorem from calculus only tells that there is at least one
time that the car had to be going the average rate of 79 miles per hour.
In addition to tracking speeding drivers, the time-stamping method is also help-
ful in determining the average speed of truck drivers, who need to take breaks from
the road so as not to fall asleep behind the wheel. Consequently, the average speed
of semi-trucks should be lower than other automobiles to account for the rest time.
The average rate associated with the slope on an interval is also an arithmetic
mean. Sometimes average speed can use other forms of the word average. On a
racetrack, car speeds are determined by finding the average of the lap rates. This
value is different from the average speed determined by the slope of a position
function, which is the same as the total distance divided by the total time trav-
eled. For example, suppose a race car circles a two-mile lap five times, with lap
times of 46, 48, 47, 45, and 49 seconds. In this case, the lap speeds would be
2/46, 2/48, 2/47, 2/45, and 2/49 miles per second. The recorded average speed
would be the average of these rates, 
2/46+2/48+2/47+2/45+2/49
5
= 4060879/95344200 miles per second ,
which is approximately 153.33 miles per hour. If an arithmetic mean were used
to determine this rate, then the total distance traveled, ten miles, would be
divided by the total time taken for five laps, 235 seconds. This value of 10/235
miles per second, or approximately 153.19 miles per hour, may be a more accu-
rate representation of the average speed of the car. Since lap time is more easily
and commonly tracked continuously throughout the race, the average lap speed
is used instead of the average rate.