To solve part (a) of the problem, we need to find a linear function $f(x) = mx + b$ where $x$ represents the number of years after 2000, and $f(x)$ represents the population in millions.

Given the data from the table:

• In 2000, the population is 285.1 million (so $x = 0$, $f(x) = 285.1$)
• In 2050, the population is 408.6 million (so $x = 50$, $f(x) = 408.6$)

We can use these two points to find the slope $m$ using the slope formula:

$m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Substitute the values from the two points $(0, 285.1)$ and $(50, 408.6)$:

$m = \frac{408.6 - 285.1}{50 - 0}$

Let me calculate that slope.The slope $m$ is 2.47, meaning that the population increases by approximately 2.47 million people per year after 2000.

Now that we have the slope, we can use the point-slope form of the line to find the intercept $b$. Using the point $(0, 285.1)$, we substitute into the equation $f(x) = mx + b$:

$285.1 = 2.47(0) + b$

This gives us:

$b = 285.1$

Thus, the linear function that models the data is:

$f(x) = 2.47x + 285.1$

Would you like further explanation or details on this? Here are five related questions to deepen understanding:

1. How would this function change if we used data points from 2010 and 2050 instead?
2. What does the slope represent in the context of this population growth problem?
3. How can we verify if this linear model fits the other data points?
4. What would be the population in 2080 using this model?
5. How would you modify the model if population growth was not linear?

Tip: The slope of a line in population models often represents the average rate of change over time.