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
282.6
2060
436.9
2010
303.7
2070
471.6
2020
332.8
2080
505.9
2030
352.6
2090
540.7
2040
383.1
2100
575.9
2050
413.4

​(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(70​). State what the value of ​f(70​) means.
​f(70​)equals
  
enter your response here ​(Round to one decimal place as​ needed.)
Part 3
State what the value of ​f(70​) means.
A.
The projected population is
  
enter your response here million in the year 2170.
B.
The projected population is
  
enter your response here million in the year 2070.
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(70)(Round to one decimal place as needed.)))State what the value of f(70) 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?

Answer 1

To solve the problem, we will first create a linear function based on the given population data from the years 2000 to 2100.

Part (a)

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

The data points we will use from the table are:

• \( (0, 282.6) \) corresponding to the year 2000
• \( (100, 575.9) \) corresponding to the year 2100

Calculating the slope \( m \):

\[ m = \frac{f(100) - f(0)}{100 - 0} = \frac{575.9 - 282.6}{100 - 0} = \frac{293.3}{100} = 2.933 \]

Calculating the intercept \( b \):

Using the point \( (0, 282.6) \): \[ f(0) = m \cdot 0 + b \implies b = 282.6 \]

Thus, the linear function that models the data is: \[ f(x) = 2.933x + 282.6 \]

Part (b)

Next, we will find \( f(70) \):

\[ f(70) = 2.933 \times 70 + 282.6 = 205.31 + 282.6 = 487.91 \]

Rounding to one decimal place: \[ f(70) \approx 487.9 \]

State what the value means:

• The value \( f(70) \) means the projected population is 487.9 million in the year 2070 (since \( 70 \) years after 2000 is the year 2070).

Part (c)

To find the population in the year 2080 using the model:

\[ f(80) = 2.933 \times 80 + 282.6 = 234.64 + 282.6 = 517.24 \]

Rounding to one decimal place: \[ f(80) \approx 517.2 \]

How does this compare with the value in the table:

The table shows the population in 2080 as 505.9 million.

Comparing the values:

• The model predicts 517.2 million, which is slightly higher than the table value of 505.9 million.

Conclusion: Thus, the answer for how this compares with the table value is: B. This value is fairly close to the table value.

Summary of Answers:

• Part (a): \( f(x) = 2.933x + 282.6 \)
• Part (b): \( f(70) \approx 487.9 \) (The projected population is 487.9 million in the year 2070)
• Part (c): The population in the year 2080 will be \( 517.2 \) million (This value is fairly close to the table value).