Population Growth and Decay

Let’s look at our warm-up example again and, this time, develop a faster method of solving the problem.

Suppose you live in a town that has a population of 1,000 people and it increasing at a rate of 5% each year. You want to find out how many people will live in your town in 5 years.

When we are working with population growth or decay there are a few pieces of information that we need to identify. As we go through our discussion, refer back to the warm-up exercise.

The first piece of information that you need is an initial population. For example, in the warm-up exercise the initial population was 1,000. We will call this value P.

Next we need to find the interest rate. For the example in the warm-up activity, the interest rate it 5% or .05. We will use this to find the value of r. It is important to remember that our value of r is 1.05 not just .05. If we look back at our warm-up we can see that we were adding the last population plus the addition for the year.

100% + 5% = 105% or 1.05

The final value that we need to know is time. In our example time is equal to 5 years. Let’s call this value t.

We can call the population after a certain number of years as A. This will give us a final general equation A = P(r)^t.

Let’s use this equation to solve our warm-up problem.

A = 1,000(1.05)⁵ ≈ 1,276

This is the same answer as we computed during our warm-up, but this is much easier. For example, if you wanted to find the population after 15 years, you would not have to work through the problem year by year, you could set up the following equation:

A = 1,000(1.05)¹⁵ ≈ 2,079.

This means that in 15 years the projected population of the town will be approximately 2,079 people.

If the population is decreasing this is called a population decay. This means that the r value will change. Instead of 100% + interest rate it would change to 100% – interest rate.