Question

The population of a city in 2008 was 32,085. Since then, the population has increased at a rate of 1.6% every 4 months. Write a function that describes the population, P(t), as a function of the number of years, t, since 2008?

Answer 1

The function to describe the population growth is P(t) = 32,085 × (1 + 0.016)^{3t}, where P(t) is the population after t years, with an initial population of 32,085 and a 1.6% growth rate applied every 4 months.

Step-by-step explanation:

To describe the population P(t) of a city as a function of the number of years t since 2008, given an initial population of 32,085 and an increase at a rate of 1.6% every 4 months, we can use the exponential growth formula:

P(t) = P0 × (1 + r)^n

Where:

• P0 is the initial population (32,085).
• r is the growth rate per period (1.6% or 0.016).
• n is the number of periods the growth rate is applied (since the rate is applied every 4 months, there are 3 periods per year).

Firstly, to convert the 4-monthly growth rate to an annual rate, remember that there are 3 periods of 4 months in a year. The function becomes:

P(t) = 32,085 × (1 + 0.016)^{3t}

This function models the population growth of the city on a yearly basis from the year 2008.