Part 1: Finding a Linear Function to Model the Data

We are given the population data for a country from 2000 to 2100. To model this data with a linear function, we'll use the formula of a line:

$f(x) = mx + b$

Where:

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

We have two points from the data that can help us determine the slope and the intercept:

• For the year 2000 ($x = 0$), the population is 276.9 million.
• For the year 2050 ($x = 50$), the population is 408.1 million.

Using these points, we can calculate the slope $m$ as follows:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{408.1 - 276.9}{50 - 0} = \frac{131.2}{50} = 2.624$

Now that we have the slope $m = 2.624$, we can substitute this value into the line equation along with one of the points to find $b$. We'll use the point $(0, 276.9)$:

$f(x) = 2.624x + 276.9$

Thus, the linear function that models the data is:

$f(x) = 2.624x + 276.9$

Part 2: Finding $f(65)$

Now, we are asked to find $f(65)$, which represents the population in the year 2065 (since $x = 65$ corresponds to 2065). Using the linear function we just derived:

$f(65) = 2.624(65) + 276.9$

$f(65) = 170.56 + 276.9 = 447.46$

Thus, the projected population in 2065 is approximately 447.5 million (rounded to one decimal place).

Part 3: Interpreting the Value of $f(65)$

Since $f(65) = 447.5$, this value represents the projected population in 2065. Therefore, the correct interpretation is:

A. The projected population is 447.5 million in the year 2065.

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Would you like any further explanation or clarification on this? Also, here are some related questions for you to explore:

1. How would you calculate the slope of a linear function from two different data points?
2. What is the significance of the intercept $b$ in a linear function modeling population growth?
3. How can we use linear models to make predictions for future years?
4. If the linear model continues indefinitely, what does the function predict about population growth in the distant future (e.g., 2200)?
5. What are the limitations of using a linear model to predict population growth?

Tip: Always check if a linear model is appropriate for the data before making predictions, especially for long-term projections, as growth may not always follow a straight line in real-world situations.