Question

the following table shows palm coast, fl as the fastest growing metropolitan area from 2000 - 2010.
use the values in the table to determine the line between the two points for palm coast, fl where ( x = ) years and ( y = ) population.
interpret the slope and ( y )-intercept.

Explanation:

Step1: Identify the two points

For Palm Coast, FL, in 2000 (let's take \( x = 2000 \)) the population \( y = 49822 \), and in 2010 ( \( x = 2010 \)) the population \( y = 95696 \). So the two points are \( (2000, 49822) \) and \( (2010, 95696) \).

Step2: Calculate the slope (\( m \))

The formula for slope is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Substituting the values: \( m=\frac{95696 - 49822}{2010 - 2000}=\frac{45874}{10}=4587.4 \).

Step3: Use point - slope form to find the equation of the line

The point - slope form is \( y - y_1=m(x - x_1) \). Using the point \( (2000, 49822) \) and \( m = 4587.4 \):
\( y-49822 = 4587.4(x - 2000) \)
Expand the right - hand side: \( y-49822=4587.4x-4587.4\times2000 \)
\( y-49822 = 4587.4x-9174800 \)
Add 49822 to both sides: \( y=4587.4x-9174800 + 49822 \)
\( y=4587.4x-9124978 \)

Step4: Interpret the slope and y - intercept

• Slope Interpretation: The slope \( m = 4587.4 \) means that the population of Palm Coast, FL is increasing at a rate of 4587.4 people per year.
• y - intercept Interpretation: The y - intercept is \( b=- 9124978 \). In the context of the equation \( y = 4587.4x + b \) (where \( x \) is the year and \( y \) is the population), when \( x = 0 \) (which is the year 0, not a meaningful year in the context of our data from 2000 - 2010), the value of \( y=-9124978 \) is not a meaningful population value in the real - world sense of the city's history, but mathematically, it is the value of \( y \) when \( x = 0 \) in the linear model.

Answer:

The equation of the line is \( y = 4587.4x-9124978 \). The slope of 4587.4 represents that the population of Palm Coast, FL increases by 4587.4 people per year. The y - intercept of - 9124978 is the value of \( y \) when \( x = 0 \) (year 0) in the linear model and is not a meaningful real - world population value for the city's history in the context of our data range.