To study the growth of a population mathematically, we use the concept of exponential models. Generally speaking, if we want to predict the increase in the population at a certain period in time, we start by considering the current population and apply an assumed annual growth rate. For example, if the U.S. population in 2008 was 301 million and the annual growth rate was 0.9%, what would be the population in the year 2050? To solve this problem, we would use the following formula:
P(1 + r)n
In this formula, P represents the initial population we are considering, r represents the annual growth rate expressed as a decimal and n is the number of years of growth. In this example, P = 301,000,000, r = 0.9% = 0.009 (remember that you must divide by 100 to convert from a percentage to a decimal), and n = 42 (the year 2050 minus the year 2008). Plugging these into the formula, we find:
P(1 + r)n = 301,000,000(1 + 0.009)42
Therefore, the U.S. population is predicted to be 438,557,000 in the year 2050.
Let’s consider the situation where we want to find out when the population will double. Let’s use this same example, but this time we want to find out when the doubling in population will occur assuming the same annual growth rate. We’ll set up the problem like the following:
Double P = P(1 + r)n
P will be 301 million, Double P will be 602 million, r = 0.009, and we will be looking for n.
Double P = P(1 + r)n
602,000,000 = 301,000,000(1 + 0.009)n
Now, we will divide both sides by 301,000,000. This will give us the following:
2 = (1.009)n
To solve for n, we need to invoke a special exponent property of logarithms. If we take the log of both sides of this equation, we can move exponent as shown below:
log 2 = log (1.009)n
log 2 = n log (1.009)
Now, divide both sides of the equation by log (1.009) to get:
n = log 2 / log (1.009)
Using the logarithm function of a calculator, this becomes:
n = log 2/log (1.009) = 77.4
Therefore, the U.S. population should double from 301 million to 602 million in 77.4 years assuming annual growth rate of 0.9 %.
Now it is your turn:
- Search the Internet and determine the most recent population of your home state. A good place to start is the U.S. Census Bureau (www.census.gov) which maintains all demographic information for the country. If possible, locate the annual growth rate for your state. If you can not locate this value, feel free to use the same value (0.9%) that we used in our example above.
- Determine the population of your state 10 years from now.
- Determine how long and in what year the population in your state may double assuming a steady annual growth rate.
- Look up the population of the city in which you live. If possible, find the annual percentage growth rate of your home city (use 0.9% if you can not locate this value).
- Determine the population of your city in 10 years.
- Determine how long until the population of your city doubles assuming a steady growth rate.
- Discuss factors that could possibly influence the growth rate of your city and state.
- Do you live in a city or state that is experiencing growth?
- Is it possible that you live in a city or state where the population is on the decline or hasn’t changed?
- How would you solve this problem if the case involved a steady decline in the population (say -0.9% annually)? Show an example.
- Think of other real world applications (besides monitoring and modeling populations) where exponential equations can be utilized.