Average Rate of Change Examples

According to Google Maps, it's 241 miles from Ann Arbor to Chicago. If Kevin made this trip in 4 hours, what was his average rate of travel during the trip?

If Kevin travelled 241 miles in 4 hours, his average rate of travel was

.

Look at a graph of Kevin's position (measured in miles from Ann Arbor) vs. time (measured in hours since the start of the trip). We don't know Kevin's exact position during most of the trip, since we don't know how fast he was driving, but we know his position was 0 miles at the beginning of the trip and 241 miles at the end of the trip:

The distance Kevin traveled is equal to his ending position minus his starting position, which can be seen on the vertical axis:

The time the trip took can be seen on the horizontal axis:

When we take the quotient

we're finding the slope of the line between two points:

Liana was 10 miles from home at 2 pm and 70 miles from home at 3:15 pm. What was her average rate of travel during the trip?

Liana traveled 70 – 10 = 60 miles over 1.25 hours. Her average rate of travel was

Again, we can graph Liana's position vs. time:

We can see that when we divide distance by time we're finding the slope of the line between these two points:

We've been saying "rate of travel" for two reasons. One reason is to avoid the words "speed" and "velocity," which are often used interchangeably but in mathematics mean different things. The other reason is that we need to be doing stuff with other rates besides miles per hour, so it's good to get in the habit of using the word "rate" correctly.

A famous author signed 200 books in two and a half hours. Find the average rate of change of the number of books signed with respect to the number of hours elapsed.

Since the number of books signed depends on how much time has elapsed, the independent variable is time (in hours) and the dependent variable is number of books signed.

The size of the time interval is 2.5 hours. The dependent variable is 0 at the start of the time interval and 200 at the end:

Now we calculate the average rate at which the author signed the books:

The phrase "on average" can be a clue that we want to find an average rate of change.

Thomas went mountain climbing and took some trail mix. After climbing 200 feet he had eaten three handfuls of trail mix. On average, how many handfuls of trail mix did he consume per foot?

The amount of trail mix Thomas has eaten depends on how far he's climbed, so the independent variable is measured in feet and the dependent variable is measured in handfuls of trail mix. We want to find the average rate of change of (handfuls of trail mix) with respect to feet. The independent variable goes from 0 ft to 200 ft.

The dependent variable goes from 0 handfuls to 3 handfuls.

The average rate of change is

Veronica has a tendency to forget about her cookies and leave them in the oven until the smoke alarm goes off. If she bakes her cookies at 300°F, it takes three-quarters of an hour for the smoke alarm to go off. If she bakes her cookies at 450°F (for those friends who like them...crispy), it only takes 10 minutes for the smoke alarm to go off. What is the average rate of change of time until the smoke alarm goes off with respect to oven temperature (in degrees Fahrenheit)?

This has some wild units: (time until the smoke alarm goes off) and (oven temperature in degrees Fahrenheit).

Abbreviate these by "time" and "temperature" respectively. Notice that in this case, time is the dependent variable because the oven temperature determines the amount of time it takes for the smoke alarm to go. Also be careful with units. The problem doesn't specify whether time should be measured in hours or minutes, so we have two choices. If we measure time in hours we find

If we measure time in minutes we find

Whatever the units, no matter how weird or how meaningless they are, the units of the average rate of change of y with respect to x are "units of y per units of x." If y is measured in cows and x is measured in catapults, then the average r.o.c. of y with respect to x is measured in "cows per catapult."