The following table gives projections of the population of a country from 2000 to 2100.
Answer parts ​(a) through ​(c).
Year
Population​ (millions)
Year
Population​ (millions)
 
2000
282.9
2060
436.8
2010
301.3
2070
471.6
2020
334.3
2080
498.9
2030
353.3
2090
539.8
2040
379.9
2100
574.8
2050
410.1
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Part 1
​(a) Find a linear function that models the​ data, with x equal to the number of years after 2000 and​ f(x) equal to the population in millions.
​f(x)equals
  
enter your response herexplus
  
enter your response here
​(Type integers or decimals rounded to three decimal places as​ needed.)

To find a linear function that models the population data, we need to determine the slope \( m \) and the y-intercept \( b \) of the line in the form:

\[
f(x) = mx + b
\]

where:
- \( x \) is the number of years after 2000, 
- \( f(x) \) is the population in millions.

We can use two points from the table to calculate the slope \( m \). Let's take the year 2000 (\( x = 0 \), \( f(x) = 282.9 \)) and the year 2100 (\( x = 100 \), \( f(x) = 574.8 \)).

### Step 1: Calculate the slope \( m \)
The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \( (0, 282.9) \) and \( (100, 574.8) \):

\[
m = \frac{574.8 - 282.9}{100 - 0} = \frac{291.9}{100} = 2.919
\]

### Step 2: Find the y-intercept \( b \)
The y-intercept is the population at \( x = 0 \), which is already given as \( f(0) = 282.9 \). So:

\[
b = 282.9
\]

### Step 3: Write the linear equation
The linear function is:

\[
f(x) = 2.919x + 282.9
\]

### Final Answer:
The linear function is:

\[
f(x) = 2.919x + 282.9
\]

Would you like further details on how the linear model works or have any questions?

Here are 5 related questions you might find useful:
1. How would the equation change if we used different years to calculate the slope?
2. How do we interpret the slope of 2.919 in this context?
3. What does the y-intercept represent in terms of population data?
4. Can this model accurately predict populations for years outside the given range?
5. How could we improve this model if the population growth is not perfectly linear?

**Tip:** When modeling with linear functions, ensure the data fits the assumption of constant growth over time to get accurate results.

The following table gives projections of the population of a country from 2000 to 2100. Find a linear function that models the data, with x equal to the number of years after 2000 and f(x) equal to the population in millions. (Type integers or decimals rounded to three decimal places as needed.)

Discover how to model population growth using a linear function. This problem involves calculating the slope and y-intercept from a dataset of population projections between 2000 and 2100. Learn how to apply the slope formula and create a linear equation to predict future populations. Suitable for students in Grades 9-11.

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