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

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 ).
(options for graphs: a, b, c, d with corresponding graph images)

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

Explanation:

Step1: Map data to x-values

First, define \( x \) as years after 2010:

• 2015: \( x=5 \), \( y=255.161 \)
• 2020: \( x=10 \), \( y=266.024 \)
• 2025: \( x=15 \), \( y=276.558 \)
• 2030: \( x=20 \), \( y=286.967 \)
• 2035: \( x=25 \), \( y=297.259 \)
• 2040: \( x=30 \), \( y=306.634 \)
• 2045: \( x=35 \), \( y=315.152 \)
• 2050: \( x=40 \), \( y=323.314 \)
• 2055: \( x=45 \), \( y=331.770 \)
• 2060: \( x=50 \), \( y=340.868 \)

Step2: Identify correct graph (part a)

Check the y-range of data: 255.161 to 340.868. Graph D has points spanning this range, matching the increasing trend and window [0,60] by [0,400].

Step3: Calculate slope for linear model

Use two points, e.g., \( (5, 255.161) \) and \( (50, 340.868) \):
Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
\( m = \frac{340.868 - 255.161}{50 - 5} = \frac{85.707}{45} \approx 1.9046 \)

Step4: Calculate y-intercept

Use point-slope form \( y - y_1 = m(x - x_1) \), substitute \( (5, 255.161) \):
\( y = 1.9046(x - 5) + 255.161 \)
\( y = 1.9046x - 9.523 + 255.161 \)
\( b = 255.161 - 9.523 = 245.638 \)

Answer:

a. D
b. \( y = 1.905x + 245.638 \)