The population of Dawnland is changing exponentially. The function [tex]f(x) = 60000(1.01)^x[/tex] models the population change. Interpret the function to describe Dawnland's population change. Explain how you know.

Question

High School

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Answer :

JamesJohnson657 JamesJohnson657 · Answer 1 · 2024-03-21T05:30:45-04:00

The function [tex]$ f(x) = 60000(1.01)^x $[/tex] models Dawnland's population change, where [tex]$ x $[/tex]represents the number of years.

Initial Population : The function begins with an initial population of 60,000, indicated by the term [tex]$ 60000 $.[/tex]
Exponential Growth : The term[tex]$ (1.01)^x $[/tex]represents exponential growth. In each year [tex]($ x $)[/tex], the population increases by a factor of 1.01, or 1%, from the previous year. This reflects continuous growth, typical in exponential models.
Interpreting Exponential Growth : Since the exponent [tex]$ x $[/tex] represents the number of years, plugging in different values of [tex]$ x $[/tex] will give us the population at those respective years. For example, when[tex]$ x = 1 $,[/tex] the population after one year is[tex]$ 60000(1.01)^1 = 60600 $,[/tex] an increase of 600 people from the initial population. Similarly, after 2 years[tex]($ x = 2 $)[/tex], the population is[tex]$ 60000(1.01)^2 = 61206 $[/tex], an increase from the previous year due to continuous growth.
Continuous Growth Rate : The growth rate of 1% is continuous, meaning the population is continuously increasing. This is different from discrete growth, where the population would increase by 1% every year in discrete steps.
Understanding the Model : The model [tex]$ f(x) = 60000(1.01)^x $[/tex]accurately reflects Dawnland's population change o[tex]$ x = 1 $[/tex]x \) increases, the population grows exponentially, reflecting the compounding effect of growth over time.