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
283.6
2060
438.8
2010
308.4
2070
467.8
2020
334.8
2080
506.5
2030
352.6
2090
541.1
2040
382.3
2100
575.2
2050
410.3
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(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 fairly close to the table value.
B.
This value is not very 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 model the population data provided, we start by acquiring a linear function in the form of \( f(x) = mx + b \), where \( x \) is the number of years after 2000.

Part 1: Find the Linear Function

We have the following population data points as (year after 2000, population):

• (0, 283.6)
• (10, 308.4)
• (20, 334.8)
• (30, 352.6)
• (40, 382.3)
• (50, 410.3)
• (60, 438.8)
• (70, 467.8)
• (80, 506.5)
• (90, 541.1)
• (100, 575.2)

To calculate the slope \( m \), we can use two points, for example, (0, 283.6) and (100, 575.2):

\[ m = \frac{f(100) - f(0)}{100 - 0} = \frac{575.2 - 283.6}{100} = \frac{291.6}{100} = 2.916 \]

Next, we use one of the points to find \( b \):

Using point (0, 283.6):

\[ f(0) = 2.916(0) + b \implies b = 283.6 \]

Therefore, the linear function modeling the population is:

\[ f(x) = 2.916x + 283.6 \]

Part 2: Find \( f(70) \)

To find \( f(70) \):

\[ f(70) = 2.916(70) + 283.6 = 204.12 + 283.6 = 487.72 \]

Rounding to one decimal place gives:

\[ f(70) = 487.7 \]

Part 3: State What \( f(70) \) Means

The value of \( f(70) = 487.7 \) means:

B. The projected population is 487.7 million in the year 2070.

Part 4: Predict the Population in 2080

To find the predicted population for 2080, we calculate \( f(80) \):

\[ f(80) = 2.916(80) + 283.6 = 233.28 + 283.6 = 516.88 \]

Rounding to one decimal place gives:

\[ f(80) = 516.9 \]

Part 5: Compare With Table Value for 2080

The value from the table for the year 2080 is 506.5 million. Our model prediction is 516.9 million.

The comparison shows:

A. This value is fairly close to the table value.

Summary of Answers:

• \( f(x) = 2.916x + 283.6 \)
• \( f(70) = 487.7 \) million
• It means B. The projected population is 487.7 million in the year 2070.
• The population in the year 2080 will be 516.9 million.
• This value is A. This value is fairly close to the table value.