Question

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
284.7
2060
440.1
2010
304.5
2070
472.4
2020
332.4
2080
505.7
2030
352.6
2090
539.3
2040
386.2
2100
572.2
2050
406.1
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
  
enter your response herexplus
  
enter your response here
​(Type integers or decimals rounded to three decimal places as​ needed.)
Part 2
​(b) Find ​f(60​). State what the value of ​f(60​) means.
​f(60​)equals
  
enter your response here ​(Round to one decimal place as​ needed.)
Part 3
State what the value of ​f(60​) means.
A.
The projected population is
  
enter your response here million in the year 2060.
B.
The projected population is
  
enter your response here million in the year 2160.
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(60)(Round to one decimal place as needed.)))State what the value of f(60) 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 start with finding a linear function that models the population based on the given data from the years 2000 to 2100.

### Part 1: Finding a Linear Function

1. **Determine Points:**
For a linear function \( f(x) = mx + b \):
- Let \( x \) be the number of years after 2000.
- Using data points for \( (0, 284.7) \) and \( (100, 572.2) \).

2. **Calculate Slope \( m \):**
\[
m = \frac{(572.2 - 284.7)}{(100 - 0)} = \frac{287.5}{100} = 2.875
\]

3. **Calculate Intercept \( b \):**
We can use one of the points to find \( b \).
Using the point \( (0, 284.7) \):
\[
f(0) = 2.875(0) + b \implies b = 284.7
\]

4. **Linear Function:**
Thus, the linear function is:
\[
f(x) = 2.875x + 284.7
\]

### Part 2: Find \( f(60) \) and Interpret

1. **Calculate \( f(60) \)**:
\[
f(60) = 2.875(60) + 284.7 = 172.5 + 284.7 = 457.2
\]
(Rounded to one decimal place, \( f(60) = 457.2 \))

2. **Interpretation**:
- The value of \( f(60) \) means:
- **A. The projected population is 457.2 million in the year 2060.**

### Part 3: Predict Population in 2080

1. **Calculate Population for 2080 \( (x = 80) \)**:
\[
f(80) = 2.875(80) + 284.7 = 230 + 284.7 = 514.7
\]
(Rounded to one decimal place, \( f(80) = 514.7 \))

### Part 4: Compare with Table Value

1. **Table Value for 2080**: 505.7 million.

2. **Comparison**:
- The value predicted by the model \( f(80) = 514.7 \) is compared to the table value \( 505.7 \).
- **Answer**: **A. This value is fairly close to the table value.**

### Summary of Answers

- **Part 1**: \( f(x) = 2.875x + 284.7 \)
- **Part 2**: \( f(60) = 457.2 \) million, meaning **the projected population is 457.2 million in the year 2060**.
- **Part 3**: The population in the year 2080 will be **514.7 million**.
- **Part 4**: This value is fairly close to the table value.