Question

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
284.1
2060
434.8
2010
303.1
2070
470.1
2020
331.9
2080
499.4
2030
353.3
2090
540.2
2040
382.6
2100
577.8
2050
409.1

​(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.)

Answer 1

To find a linear function that models the population data, we need to determine a linear equation of the form:

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

where \( x \) is the number of years after 2000, \( f(x) \) is the population in millions, \( m \) is the slope of the line, and \( b \) is the y-intercept.

### Step 1: Define the Points

We will use two points to calculate the slope \( m \). Let’s take the years 2000 and 2100:

- For 2000 (which corresponds to \( x = 0 \)): Population = 284.1 million
- For 2100 (which corresponds to \( x = 100 \)): Population = 577.8 million

The points are:
- \( (0, 284.1) \)
- \( (100, 577.8) \)

### Step 2: Calculate the Slope \( m \)

The formula for calculating the slope \( m \) is given by:

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

Using our points:

\[
m = \frac{577.8 - 284.1}{100 - 0} = \frac{293.7}{100} = 2.937
\]

### Step 3: Calculate the Y-Intercept \( b \)

Using the point \( (0, 284.1) \) to find \( b \):

\[
f(0) = 2.937 \cdot 0 + b = 284.1
\]
\[
b = 284.1
\]

### Final Function

Putting it all together, the linear function that models the population is:

\[
f(x) = 2.937x + 284.1
\]

### Final Answer

Therefore, the function is:

- **Slope**: \( 2.937 \)
- **Y-intercept**: \( 284.1 \)

So, the final answer is:

\[ f(x) = 2.937x + 284.1 \]

(rounded to three decimal places as needed).