Question

the table gives the projections of the population, in millions, of a countrys residents over age 16 for selected years from 2015 to 2060. answer parts (a) through (e). the linear model is a very good fit for the data. d. what does the model predict that the population will be in 2042?

Explanation:

Step1: Assume linear model $y = mx + b$. Let $x$ be the number of years since 2015. So for 2015, $x = 0$; for 2020, $x=5$ etc.

We can use two - point form to find the equation of the line. Let's take two points $(x_1,y_1)=(0,255.161)$ and $(x_2,y_2)=(45,340.868)$ (corresponding to 2015 and 2060).
The slope $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{340.868 - 255.161}{45}=\frac{85.707}{45}\approx1.9046$.
Since when $x = 0$, $y = 255.161$, the $y$-intercept $b = 255.161$. So the linear model is $y=1.9046x + 255.161$.

Step2: For the year 2042, $x$ is the number of years since 2015. So $x=2042 - 2015=27$.

Substitute $x = 27$ into the linear model $y=1.9046\times27+255.161$.
$y = 1.9046\times27+255.161=51.4242+255.161 = 306.5852$.

Answer:

306.5852