# In the Billions and Exponential Modeling

Alignments to Content Standards: F-LE.A.1.c

The data in the table below was taken from Wikipedia.

Year World Population in Billions (Estimate)
1804 1
1927 2
1960 3
1974 4
1987 5
1999 6
2012 7
1. For each span of years in the table below, assume that the relationship between the population, $P$, and the number of years since the beginning of the time period, $t$, is exponential and then determine the annual rate of growth $r$ for that range of years.

For example, for the range of years $1804$ through $1927$, we have $P(t) = 1,000,000,000 \cdot b^t$, assumine an an exponential relationship. Since $1927$ is $123$ years after $1804$, the population in $1927$ can be expressed as $P(123)$ and we have $$P(123) = 1,000,000,000 \cdot b^{123} = 2,000,000,000.$$ This means $$b = \sqrt[123]{2} \approx 1.006$$ or that the population grew at a rate of approximately $1.006 - 1 = 0.006$ or $0.6\%$ for each year between 1804 and 1927.

Span of Years Approximate Annual World Population Growth Rate $r$
1804 - 1927 0.6%
1927 - 1960
1960 - 1974
1974 - 1987
1987 - 1999
1999 - 2012
2. How many times bigger is the growth rate from 1927 to 1960 than the growth rate from 1804 to 1927?
3. Based on your answers to parts (a) and (b) would an exponential function be appropriate to model the relationship between the world population and the year? Explain why or why not.
4. Brainstorm some possible explanations for the overall behavior of the growth rates in part (a).

## IM Commentary

This problem provides an opportunity to experiment with modeling real data. Populations are often modeled with exponential functions and in this particular case we see that, over the last $200$ years, the rate of population growth accelerated rapidly, reaching a peak a little after the middle of the $20^{\rm th}$ century and now it is slowing down. Although an exponential model is not accurate for this long period, finding and comparing the growth rates at different times provides valuable insight into the recent behavior of the human population.

This problem assumes students have completed several preliminary tasks about the fact that exponential functions change by equal factors over equal intervals (e.g., F-LE US Population 1790-1860 and F-LE Basketball Rebounds).

## Solution

1. Employing the solution method illustrated in part a) of the problem we have:
 Span of Years Approximate Annual World Population Growth Rate $r$ 1804 - 1927 $0.006 = 0.6\%$ 1927 - 1960 $0.012 = 1.2\%$ 1960 - 1974 $0.021 = 2.1 \%$ 1974 - 1987 $0.017 = 1.7\%$ 1987 - 1999 $0.015 = 1.5\%$ 1999 - 2012 $0.012 = 1.2\%$
2. The growth rate from 1927 to 1960 ($1.2\%$) is two times the growth rate from 1804 to 1927 ($0.6\%$).
3. An exponential model is appropriate when the annual rate of population growth (or decay) does not vary, or shows very little variation. In this case, the average annual rate of population changes significantly over this two hundred year period. It increases at first, then remains fairly steady, and finally decreases slightly over the last 38 years. Because of this, an exponential model is inappropriate for this data.
4. One possible explanation for why the average rate of growth for the population increased in the middle of the twentieth century and is now decreasing has to do with resources. The industrial revolution provided a wealth of natural resources and the means of transporting these resources, thereby allowing populations to grow rapidly. As we have moved further into the twentieth and twenty-first centuries, however, resources such as clean water are becoming scarce and the natural impact of this is to decrease the overall population growth rate.