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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Exponential Growth and Population Projections: A Case Study of Illinois and Northbrook,
IL
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
= 301,000,000(1.009)42
= 301,000,000(1.457)
= 438,557,000
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)
Exponential Growth and Population Projections: A Case Study of Illinois and Northbrook,
IL
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
= 301,000,000(1.009)42
= 301,000,000(1.457)
= 438,557,000
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)
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Subject
Mathematics