Question

At the beginning of a population study, a city had 300,000 people. Each year since, the population has grown by 2.5%.

Let t be the number of years since start of the study. Let y be the city's population.
Write an exponential function showing the relationship between y and t.

Answer 1

The population growth can be modeled by an exponential function of the form:

y = ab^t

where y is the population, t is the number of years since the start of the study, a is the initial population, and b is the annual growth rate expressed as a decimal.

In this case, the initial population a is 300,000, and the annual growth rate is 2.5%, or 0.025 as a decimal. Therefore, the exponential function is:

y = 300,000 * (1 + 0.025)^t

Simplifying the expression inside the parentheses, we get:

y = 300,000 * 1.025^t

This is the exponential function that shows the relationship between the city's population y and the number of years t since the start of the study.