Question

The population of a town is 6500 and is increasing at a rate of 4% each year. Create a function for this scenario. At this rate, approximately what will the town's population be in 4 years?

Answer 1

Answer:

$P_4 = 7604.08$

Step-by-step explanation:

If the population increases at a rate of 4% per annum, then:

In year 1:

$P_1 = P_0 + 0.04P_0$

Where $P_0$ is the initial population and $P_n$ is the population in year n

In year 2

$P_2 = P_1 + 0.04P_1$

It can also be written as:

$P_2 = P_0 + 0.04P_0 + 0.04 (P_0 + 0.04P_0)$

Taking out common factor $P_0$

$P_2 = (1 + 0.04) (P_0) + 0.04P_0 (1 + 0.04)$

Taking out common factor (1 + 0.04)

$P_2 = (1 + 0.04) (P_0 + 0.04P_0)$

Taking out again common factor $P_0$

$P_2 = (1 + 0.04) (1 + 0.04) P_0$

Simplifying

$P_2 = P_0 (1 + 0.04) ^ 2$

So

$P_n = P_0 (1 + 0.04) ^ n$

This is the equation that represents the population for year n

Then, in 4 years, the population will be:

$P_4 = P_0 (1 + 0.04) ^ 4\\P_4 = 6500(1 + 0.04) ^ 4\\P_4 = 7604.08$