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
Question content area bottom
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.

Answer 1

To find a linear function that models the population data, we first need to determine two points that we can use to formulate a linear equation of the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

From the provided data, we can choose two points. For this example, let’s use the years 2000 and 2100:

1. For the year 2000 (\( x = 0 \)):
- Population = 284.1 million → point (0, 284.1)

2. For the year 2100 (\( x = 100 \)):
- Population = 577.8 million → point (100, 577.8)

### Step 1: Calculate the slope \( m \)

The slope \( m \) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the points we have:
\[
m = \frac{577.8 - 284.1}{100 - 0} = \frac{293.7}{100} = 2.937
\]

### Step 2: Find the y-intercept \( b \)

Now we can use one of the points to calculate \( b \). We'll use the point (0, 284.1):
\[
f(0) = m(0) + b \implies 284.1 = 2.937(0) + b \implies b = 284.1
\]

### Step 3: Write the linear function

Now we can write the linear function that models the population:
\[
f(x) = 2.937x + 284.1
\]

### Conclusion

The linear function that models the population of the country based on the data provided is:
\[
f(x) = 2.937x + 284.1
\]
where \( x \) is the number of years after 2000, and \( f(x) \) is the population in millions.