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).

year	population (millions)	year	population (millions)
2020	266.024	2045	315.152
2025	276.558	2050	323.314
2030	286.967	2055	331.770
2035	297.259	2060	340.868

all graphs have viewing window 0, 60 by 0, 400 with xscl = 5 and yscl = 50.
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: Let \(x = 0\) represent the year 2015. Then for the year 2042, \(x=2042 - 2015=27\).

Since the linear - model is a very good fit for the data, assume the linear model is of the form \(y = mx + b\). We can use two points \((x_1,y_1)\) and \((x_2,y_2)\) from the data to find the equation of the line. Let's take \((x_1 = 0,y_1=255.161)\) (corresponding to 2015) and \((x_2 = 25,y_2 = 297.259)\) (corresponding to 2040).
First, find the slope \(m\) using the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
\[m=\frac{297.259 - 255.161}{25-0}=\frac{42.098}{25}=1.68392\]
Since \(y_1 = 255.161\) and \(x_1 = 0\), the \(y\) - intercept \(b = 255.161\). So the linear model is \(y=1.68392x + 255.161\).

Step2: Substitute \(x = 27\) into the linear model.

\[y=1.68392\times27+255.161\]
\[y = 45.46584+255.161\]
\[y=300.62684\]

Answer:

300.627 (rounded to three decimal places)