Answer :
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.
