Question

# The population of a small town was 3,600 people in the year 2015. The population increases by 4.5% every year. Write the exponential equation that models the population of the town t years after 2015. Then use your equation to estimate the population in the year 2025?

Exponential models
The population of a small town was 3,600 people in the year 2015. The population increases by 4.5% every year.
Write the exponential equation that models the population of the town t years after 2015.
Then use your equation to estimate the population in the year 2025?

2021-02-10

The general form of exponential growth is
$$\displaystyle{y}={y}_{{0}}{e}^{{{k}{t}}}$$
where,
$$\displaystyle{y}_{{0}}=$$ initial population
k= constant growth rate
In this question, we have been given the initial population and a constant growth rate which is
$$\displaystyle{y}_{{0}}={3600}$$
$$k=4.5\%=0.045$$
Substituting the values in equation (1) we have the equation that models the population of the town t years after 2015
$$\displaystyle{y}={3600}{e}^{{{0.045}{t}}}$$
Here we have to calculate the population in 2025 which means the time $$t = 10$$ after 2015
$$\displaystyle{y}={3600}{e}^{{{0.045}{\left({10}\right)}}}$$
$$\displaystyle{y}={3600}{e}^{{{0.45}}}$$
$$\displaystyle{y}={3600}\times{1.56}$$
$$\displaystyle{y}={5647.49}$$
$$\displaystyle{y}={5648}$$(approx)
hence, the population in 2025 is $$y=5648$$