A town had a population of 65,000 in 2000.

Question

15. A town had a population of 65,000 in 2000. Then the population increased by 2.5% each year for the next 5 years. a. Write a function that models the population over time since the year 2000. b. Use the function to predict the population in 2004

Answer 1

Hello, Paola,

Imagine a table with years in one column and population in the next. The first row starts with the year 2000 and population 65000. This population is our "base." Let's establish Y_n as the population for the nth year after 2000. For example, Y₀ is year 2000, Y₁ is 2001, Y₂ is 2002, and so on. [n = 0, 1, 2, 3, 4, . . . .]

We can see a pattern if we look at what happens the first two years:

Y₀ = (65000)

Y₁ = (65000)*(1+0.025) The 1.025 represents the 65,000 we started with and the 0.025 is the increase for the next year. This gives us a population of 66625 for Y₁, or 2001.

I'll use Y₀ instead of 65000 in the next steps. Y₀ is defined by the problem, but we can use any starting value to get a correct answer to what that value would be in n incremental years.

If we continue to the next year. We take the population of 2001 and increase it again by (1+0.025).

Y₂ = Y₁*(1.025) = 68291 for 2002 [Let's round to the nearest whole number, since fractions of a person are only allowed in movies.]

If we did each year in the same manner, we'll see a repeating trend in the numbers. That trend can be expressed by:

Y_n = (65000)*(1.025)ⁿ

In other words, we can take the starting population (Y₀) and multiply it by(1.025) n times to get the population for that year.

Y₃ = Y1(1.025)(1.025)(1.025)

Year (2000+3) has a population of (65000)*(1.025)ⁿ

2003 population = 69998

The general equation can be stated as Y_n = Y₁*(1.025)ⁿ

Bob