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Part 1
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
276.9
2060
435.4
2010
306.9
2070
467.8
2020
332.4
2080
505.2
2030
359.7
2090
534.6
2040
380.1
2100
578.3
2050
408.1
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.
​f(x)equals
  
2.916xplus
  
271.050
​(Type integers or decimals rounded to three decimal places as​ needed.)
Part 2
​(b) Find ​f(65​). State what the value of ​f(65​) means.
​f(65​)equals
  
460.6 ​(Round to one decimal place as​ needed.)
Part 3
State what the value of ​f(65​) means.
A.
The projected population is 
  
enter your response here million in the year 2065.
B.
The projected population is 
  
enter your response here million in the year 2165.

### 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**.

---

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.

The following table gives projections of the population of a country from 2000 to 2100. Answer parts ​(a) through ​(c). Year Population​ (millions) 2000 276.9 2010 306.9 2020 332.4 2030 359.7 2040 380.1 2050 408.1 2060 435.4 2070 467.8 2080 505.2 2090 534.6 2100 578.3. 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 2.916xplus 271.050. Part 2: ​(b) Find ​f(65​). State what the value of ​f(65​) means. f(65​)equals 460.6. Part 3: State what the value of ​f(65​) means. A. The projected population is enter your response here million in the year 2065. B. The projected population is enter your response here million in the year 2165.

Learn how to model population projections using a linear function. This example involves calculating the slope and intercept, finding specific population values, and interpreting the results. Suitable for high school students (Grades 9-12).

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