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he 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
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?
A.
This value is not very close to the table value.
B.
This value is fairly close to the table value.
f(x)xx(Type integers or decimals rounded to three decimal places as needed.)f(75)(Round to one decimal place as needed.)))State what the value of f(75) means.The population in the year 2080 will be(Round to one decimal place as needed.)How does this compare with the value for 2080 in the table?

1 answer

To find a linear model that approximates the population data of the country from 2000 to 2100 based on the given table, we'll first establish the years and corresponding population values.

Data Summary

| Year | Population (millions) | |------|-----------------------| | 2000 | 281.6 | | 2010 | 306.8 | | 2020 | 330.8 | | 2030 | 357.3 | | 2040 | 386.5 | | 2050 | 411.8 | | 2060 | 437.8 | | 2070 | 470.8 | | 2080 | 502.8 | | 2090 | 541.8 | | 2100 | 574.9 |

Part 1: Finding a Linear Function f(x)

Let \( x \) represent the number of years after 2000. The year 2000 corresponds to \( x = 0 \), and so on.

We can fit a linear function of the form: \[ f(x) = mx + b \]

To find the slope (\( m \)) and y-intercept (\( b \)), we can use two points from the data table. A good choice is the values for the years 2000 (\( x = 0, f(0) = 281.6 \)) and 2100 (\( x = 100, f(100) = 574.9 \)).

  1. Calculate the slope \( m \): \[ m = \frac{f(100) - f(0)}{100 - 0} = \frac{574.9 - 281.6}{100} = \frac{293.3}{100} = 2.933 \]

  2. Calculate the y-intercept \( b \) using \( f(0) = 281.6 \): \[ b = 281.6 \]

Thus, the linear function is: \[ f(x) = 2.933x + 281.6 \]

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

Using the linear function to find \( f(75) \): \[ f(75) = 2.933(75) + 281.6
\] Calculating: \[ f(75) = 219.975 + 281.6 = 501.575 \]

Rounded to one decimal place: \[ f(75) \approx 501.6 \]

Part 3: State What the Value of \( f(75) \) Means

Since \( 75 \) corresponds to the year \( 2000 + 75 = 2075 \): B. The projected population is 501.6 million in the year 2075.

Part 4: What Does This Model Predict the Population to Be in 2080?

Calculating \( f(80) \): \[ f(80) = 2.933(80) + 281.6 \] Calculating: \[ f(80) = 234.64 + 281.6 = 516.24 \]

Rounded to one decimal place: \[ f(80) \approx 516.2 \]

Part 5: How Does This Compare with the Value for 2080 in the Table?

Table value for 2080 is 502.8 million. The predicted value from the model is 516.2 million.

Comparison: B. This value is fairly close to the table value.

Final Answers:

  1. \( f(x) = 2.933x + 281.6 \)
  2. \( f(75) = 501.6 \) million
  3. The projected population is 501.6 million in the year 2075.
  4. The population in the year 2080 will be 516.2 million.
  5. This value is fairly close to the table value.