To model the data using a linear function, we need to find the slope and the intercept of the line [tex]\( f(x) = mx + b \)[/tex], where [tex]\( x \)[/tex] is the number of years after 2000 and [tex]\( f(x) \)[/tex] is the population in millions.
Given the data points:
- Year 2000 (which corresponds to [tex]\( x = 0 \)[/tex]) has a population of 2795 million.
- Year 2010 ([tex]\( x = 10 \)[/tex]) has a population of 3059 million.
- Year 2020 ([tex]\( x = 20 \)[/tex]) has a population of 3348 million.
- Year 2030 ([tex]\( x = 30 \)[/tex]) has a population of 3568 million.
- Year 2040 ([tex]\( x = 40 \)[/tex]) has a population of 3829 million.
- Year 2050 ([tex]\( x = 50 \)[/tex]) has a population of 4134 million.
- Year 2060 ([tex]\( x = 60 \)[/tex]) has a population of 4375 million.
- Year 2070 ([tex]\( x = 70 \)[/tex]) has a population of 4652 million.
- Year 2080 ([tex]\( x = 80 \)[/tex]) has a population of 5056 million.
- Year 2100 ([tex]\( x = 100 \)[/tex]) has a population of 5749 million.
By performing linear regression on these data points, we find that the slope [tex]\( m \)[/tex] and the intercept [tex]\( b \)[/tex] are:
[tex]\[ m \approx 28.786 \, \text{(rounded to three decimal places)} \][/tex]
[tex]\[ b \approx 2732.357 \, \text{(rounded to three decimal places)} \][/tex]
Therefore, the linear function that models the data is:
[tex]\[ f(x) = 28.786x + 2732.357 \][/tex]
So the function [tex]\( f(x) \)[/tex] that models the population in millions [tex]\( x \)[/tex] years after 2000 is:
[tex]\[ f(x) = 28.786x + 2732.357 \][/tex]
This function can now be used to estimate the population at any given year after 2000.
