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Exponential Growth and Population Projections: A Case Study of Illinois and Northbrook, IL

A mathematical assignment applying exponential growth models to predict population trends in Illinois.

Ethan Wilson
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1Exponential Growth and Population Projections: A Case Study of Illinois and Northbrook,ILTo 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 intime, 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 was0.9%, what would be the population in the year 2050? To solve this problem, we would use thefollowing formula:P(1 + r)nIn this formula, P represents the initial population we are considering, r represents the annualgrowth 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 apercentage to a decimal), and n = 42 (the year 2050 minus the year 2008). Plugging these intothe formula, we find:P(1 + r)n = 301,000,000(1 + 0.009)42= 301,000,000(1.009)42= 301,000,000(1.457)= 438,557,000Therefore, 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 usethis same example, but this time we want to find out when the doubling in population will occurassuming the same annual growth rate. We’ll set up the problem like the following:Double P = P(1 + r)nP 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)n602,000,000 = 301,000,000(1 + 0.009)nNow, we will divide both sides by 301,000,000. This will give us the following:2 = (1.009)nTo solve for n, we need to invoke a special exponent property of logarithms. If we take the log ofboth sides of this equation, we can move exponent as shown below:log 2 = log (1.009)nlog 2 = n log (1.009)

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