Linear Model for the Situation:
To find the linear model for the situation, we can use the two data points provided: in 2007, there were 130 lights, and in 2011, there were 146 lights. We can use these points to determine the equation of the line representing the growth of streetlights over time.
Let's denote the year as x and the number of streetlights as y. We can create two equations based on the given data:
Equation 1: 2007 → x = 0, y = 130 Equation 2: 2011 → x = 4, y = 146
Now we can use these two points to find the slope (m) and y-intercept (b) of the linear equation in the form y = mx + b.
First, let's calculate the slope (m): m = (y2 - y1) / (x2 - x1) m = (146 - 130) / (4 - 0) m = 16 / 4 m = 4
The slope of the line is 4.
Next, let's calculate the y-intercept (b) using one of the points (2007, 130): 130 = 4 * 0 + b b = 130
The y-intercept of the line is 130.
Therefore, the linear model for this situation is: y = 4x + 130
Predicting the Number of Lights This Year:
To predict how many lights there are this year, we need to determine the value of y when x represents the current year.
Let's denote the current year as x = 2021.
Using the linear model equation, we can substitute x = 2021 to find y: y = 4 * 2021 + 130 y = 8084 + 130 y = 8214
Therefore, it is predicted that there are 8214 lights in Darksville this year.
Determining When There Will Be 300 Lights:
To determine when there will be 300 lights, we need to find the value of x when y represents 300.
Using the linear model equation, we can substitute y = 300 and solve for x: 300 = 4x + 130 4x = 300 - 130 4x = 170 x = 170 / 4 x ≈ 42.5
Therefore, there will be approximately 300 lights in Darksville around the year 2042.5 (between 2042 and 2043).
Please note that predicting future values based on a linear model assumes that the growth pattern will continue to be linear, which may not always be the case in real-world situations.