Peggy drove from New York to Boston (a distance of 219 miles) in 4 hours. What was her average rate of travel?
Answer
Peggy's average rate is given by
.
Example 2
Lou was 20 miles from home at 1 pm and 120 miles from home at 3 pm. What was Lou's average rate of travel during that time?
Answer
Since Lou started 20 miles from home and ended 120 miles from home, he drove 120 – 20 = 100 miles over 2 hours.
Example 3
The graph shows John's starting and ending distances from home on a recent trip. Calculate his average rate of travel during the trip.
Answer
From the picture, we can find the distance and time:
Example 4
The graph shows Suhaila's position during a trip. Calculate Suhaila's average rate of travel between 2 pm and 2:30 pm.
Answer
From the picture, we can find the distance and time:
The "rate of travel" we've been talking about is an example of an average rate of change, abbreviated "average r.o.c.". When we talk about "rate of travel," we're talking about the average rate of change of distance traveled with respect to time. Similarly, if x is any independent variable (not necessarily time) and y is a dependent variable (not necessarily miles) related to x, we can find the average r.o.c. of y with respect to x on a given interval.
Example 5
Roger likes to eat jellybeans. One day he ate 56 jellybeans in 7 minutes. On average, how many jellybeans did he eat per minute?
Answer
The average number of jellybeans Roger ate per minute was
Example 6
When there are two mome raths in the forest, there are two borogoves. When there are twenty mome raths in the forest, there are 15 borogoves. What is the average rate of change of borogoves with respect to
mome raths?
Answer
The units here are meaningless, but we can still find the requested rate of change. Since the problem asks for the average r.o.c. of borogoves with respect to mome raths, "borogoves" is the dependent variable and "mome raths" are the independent variable. The change in mome raths is 20 – 2 = 18, and the change in borogoves is 13 – 2 = 11. Therefore
Example 7
An army is using catapults against their enemy. They go through the catapults one at a time, tossing them aside when they break. After the first catapult breaks they've thrown 20 stones. After the 19th catapult breaks, they've thrown 220 stones. On average, how many stones does each catapult throw?
Answer
We're looking for the average r.o.c. of stones with respect to catapults.
If we consider going from 1 to 19 catapults, the change in number of stones thrown is 220 – 20 = 200, and the change in number of catapults used is 19 – 1 = 18.
The average r.o.c. is
Example 8
Alice went to Wonderland and visited a succession of tea parties given by the Mad Hatter. The number of dormice at the tea parties changed depending on the number of teapots laid out.
A party with 3 teapots would have only one dormouse, but a party with 14 teapots would have 24 dormice. What was the average rate of change of dormice with respect to teapots?
Answer
This is another one with weird units. The independent variable is teapots, and the dependent variable is dormice. Therefore
.
