Session 7, Part B:
Exponential Growth

In This Part: A Salary Situation | Population Growth

Many populations -- human, plant, and bacteria -- grow exponentially, at least at first. In time, these populations start to lose their resources (space, food, and so on).
Note 6

Here's an example:

"Whale Numbers up 12% a Year" was a headline in a 1993 Australian newspaper. A 13-year study had found that the humpback whale population off the coast of Australia was increasing significantly. The actual data suggested the increase was closer to 14 percent!

When the study began in 1981, the humpback whale population was 350. Suppose the population has been increasing by about 14 percent each year since then. To find an increase of 14 percent, you could do either of the following (P stands for population):

Pthis year = Plast year + (0.14) * Plast year

Pthis year = (1.14) * Plast year

Each of these is a recursive rule. The first rule says that to know this year's population, start with last year's population (which is Plast year), then add the population growth. Since the growth is 14 percent of last year's population,

The second rule is used more frequently because it's easier to calculate -- it incorporates the adding in one calculation. This equation shows that the second computation is equivalent to the first:

x + (0.14) * x = (1.14) * x

The second computation fits the format of an exponential function, because successive outputs have the same ratio (in this situation, the ratio is 1.14).

Problem B4

Make a table that shows the estimated whale population for the 5 years after 1981.
Note 7

 Years after 1981 Estimated Population 0 350 1 2 3 4 5

 Years after 1981 Estimated Population 0 350 1 399 2 455 3 519 4 591 5 674

 Problem B5 If the whale population continues to grow by 14 percent per year, predict how many whales there would be in 2001 (20 years after 1981).

 See if you can do this problem without extending your table from Problem B4. You would need to use a closed-form rule to "jump" directly to the 2001 answer.   Close Tip Think of a pattern that would require you to double the number of toothpicks you use at each step. That would form an exponential function, because successive outputs have a common ratio (2).

 Problem B6 How many years does it take the whale population to double if it grows at this rate? Does your answer depend on the starting value of the whale population?

 Video Segment In this video segment, taken from the "real world" example at the end of the Session 7 video, Mary Bachman of the Harvard School of Public Health discusses the exponential model of population growth, the factors that affect the model, and the uses of population modeling. Do you think the whale population discussed in Problems B3-B6 could increase at 14 percent per year forever? Why or why not? You can find this segment on the session video, approximately 22 minutes and 40 seconds after the Annenberg Media logo.

 Problem B7 Look at the following toothpick pattern. The number of toothpicks needed to build each stage of the pattern is a linear function. Note 8 Create a toothpick pattern in which the number of toothpicks you need for each stage is an exponential function.

 Think of a pattern that would require you to double the number of toothpicks you use at each step. That would form an exponential function, because successive outputs have a common ratio (2).   Close Tip Think of a pattern that would require you to double the number of toothpicks you use at each step. That would form an exponential function, because successive outputs have a common ratio (2).

 The "Population Growth" problem taken from IMPACT Mathematics Course 2, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p.186. www.glencoe.com/sec/math

Next > Part C: Figurate Numbers

 Session 7: Index | Notes | Solutions | Video