The following table gives projections of the population of a country from 2000 to 2100.

Question

Answer parts ​(a) through ​(c).
Year
Population​ (millions)
Year
Population​ (millions)

2000
281.6
2060
437.8
2010
306.8
2070
470.8
2020
330.8
2080
502.8
2030
357.3
2090
541.8
2040
386.5
2100
574.9
2050
411.8
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
  
enter your response herexplus
  
enter your response here
​(Type integers or decimals rounded to three decimal places as​ needed.)
Part 2
​(b) Find ​f(75​). State what the value of ​f(75​) means.
​f(75​)equals
  
enter your response here ​(Round to one decimal place as​ needed.)
Part 3
State what the value of ​f(75​) means.
A.
The projected population is
  
enter your response here million in the year 2175.
B.
The projected population is
  
enter your response here million in the year 2075.
Part 4
​(c) What does this model predict the population to be in​ 2080? How does this compare with the value for 2080 in the​ table?
The population in the year 2080 will be
  
enter your response here million.
​(Round to one decimal place as​ needed.)
Part 5
How does this compare with the value for 2080 in the​ table?

Answer 1

To answer your questions, we need to find a linear function that models the population data given, and then evaluate it based on the required parts.

Part 1: Finding the linear function

We will create a linear function \( f(x) = mx + b \), where \( x \) is the number of years after 2000, and \( f(x) \) is the population in millions.

First, we extract the data points from the table:

• (0, 281.6)
• (60, 437.8)
• (100, 574.9)

To find the slope \( m \):

\[ m = \frac{f(100) - f(0)}{100 - 0} = \frac{574.9 - 281.6}{100 - 0} = \frac{293.3}{100} = 2.933 \]

Now, using the point-slope form for one of the points (let's use (0, 281.6)):

• \( b \) is the population at the year 2000, which is 281.6 million.

Thus, the linear equation is:

\[ f(x) = 2.933x + 281.6 \]

So the linear function \( f(x) \) is:

1. a) \( f(x) = 2.933x + 281.6 \)

Part 2: Finding \( f(75) \)

Now we substitute \( x = 75 \):

\[ f(75) = 2.933(75) + 281.6 \]

Calculating this:

\[ f(75) = 219.975 + 281.6 = 501.575 \]

Rounding to one decimal place gives us:

2. b) \( f(75) = 501.6 \)

Part 3: Interpretation of \( f(75) \)

• \( f(75) \) corresponds to the year \( 2000 + 75 = 2075 \).

So the correct interpretation is:

3. Answer: B. The projected population is 501.6 million in the year 2075.

Part 4: Predicting the population in 2080

Next, we predict \( f(80) \) (2080 corresponds to \( x = 80 \)):

\[ f(80) = 2.933(80) + 281.6 \]

Calculating this:

\[ f(80) = 234.64 + 281.6 = 516.24 \]

Rounding to one decimal place gives:

4. Answer: The population in the year 2080 will be 516.2 million.

Part 5: Comparison with the table

The table shows that the population in 2080 is 502.8 million. Our model predicts:

• \( 516.2 \) million (model prediction)
• \( 502.8 \) million (from the table)

5. Comparison: The model prediction of 516.2 million is higher than the table value of 502.8 million.

Summary of Answers

1. (a) \( f(x) = 2.933x + 281.6 \)
2. (b) \( f(75) = 501.6 \)
3. (c) B. The projected population is 501.6 million in the year 2075.
4. (d) The population in the year 2080 will be 516.2 million.
5. (e) The model prediction is higher (516.2 million) than the table value (502.8 million).