To find a linear function that models the population data given, we will first consider the provided data points:
- (0, 276.9) for the year 2000
- (60, 433.5) for the year 2060
- (80, 503.9) for the year 2080
- and so forth...
To derive a linear function of the form \( f(x) = mx + b \), we need to:
-
Find the slope \( m \) using two known points. We will use the points for 2000 (x = 0, population = 276.9) and 2060 (x = 60, population = 433.5):
\[ m = \frac{f(60) - f(0)}{60 - 0} = \frac{433.5 - 276.9}{60 - 0} = \frac{156.6}{60} \approx 2.610 \]
-
Next, we use one of the points, say (0, 276.9), to find \( b \):
\[ f(0) = m \cdot 0 + b \implies 276.9 = b \]
Thus, our linear function is:
\[ f(x) = 2.610x + 276.9 \]
Part 1: The function is:
\[ f(x) = 2.610x + 276.900 \]
Part 2: To find \( f(60) \):
\[ f(60) = 2.610 \cdot 60 + 276.9 \approx 156.6 + 276.9 = 433.5 \]
So,
\[ f(60) = 433.5 \]
State what the value of \( f(60) \) means:
A. The projected population is 433.5 million in the year 2060.
Part 4: To find the population in 2080:
\[ f(80) = 2.610 \cdot 80 + 276.9 \approx 208.8 + 276.9 = 485.7 \]
So,
\[ \text{The population in the year 2080 will be } 485.7 \]
How does this compare with the value for 2080 in the table?
The value in the table is 503.9 million.
Since 485.7 million is not very close to the table value 503.9 million,
B. This value is not very close to the table value.
Final Summary Responses:
Part 1: \[ f(x) = 2.610x + 276.900 \]
Part 2: \[ f(60) = 433.5 \]
Meaning: A. The projected population is 433.5 million in the year 2060.
Part 4: \[ \text{The population in the year 2080 will be } 485.7 \]
Comparison: B. This value is not very close to the table value.