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	340.368

a. graph the data, with ( y ) equal to the population in millions and ( x ) equal to the number of years after 2010. choose the correct graph below.

all graphs have viewing window (0, 60) by (0, 400) with ( \text{xscl} = 5 ) and ( \text{yscl} = 50 ).

a. graph a
b. graph b
c. graph c
d. graph d

b. find a linear function that models the data, with ( y ) equal to the population in millions and ( x ) equal to the number of years after 2010.

( y = square x + square )

(type integers or decimals rounded to three decimal places as needed.)

Explanation:

Step1: Map x to years after 2010

For 2015: $x=5$, $y=255.161$
For 2020: $x=10$, $y=266.024$
For 2025: $x=15$, $y=276.558$
For 2030: $x=20$, $y=286.967$
For 2035: $x=25$, $y=297.259$
For 2040: $x=30$, $y=306.634$
For 2045: $x=35$, $y=315.152$
For 2050: $x=40$, $y=323.314$
For 2055: $x=45$, $y=331.770$
For 2060: $x=50$, $y=340.368$

Step2: Identify correct graph (part a)

Check y-values: all are 255-340, within [0,400]. x-values 5-50, within [0,60]. The points show a steady upward trend, matching Graph A (the top graph with increasing points in the correct range).

Step3: Calculate slope for linear model

Use first and last points: $(x_1,y_1)=(5,255.161)$, $(x_2,y_2)=(50,340.368)$
Slope $m=\frac{y_2-y_1}{x_2-x_1}=\frac{340.368-255.161}{50-5}=\frac{85.207}{45}\approx1.893$

Step4: Calculate y-intercept

Use $y=mx+b$, substitute $x=5$, $y=255.161$, $m=1.893$
$255.161=1.893\times5+b$
$255.161=9.465+b$
$b=255.161-9.465=245.696$

Answer:

a. A
b. $y=1.893x + 245.696$