Question

1. an arctic cold front is moving through an area. it is 37° when the temperature begins to drop. the scatter plot suggests a linear relationship between the temperature and the number of hours since the cold front arrived.

a. what does the rate of change, or slope, represent in this situation?
b. what is the y-intercept for the trend line and what does it represent?
c. what equation relates the change in temperature, y, to the number of hours after the cold front arrives, x?
(there is a chart with temperature (°f) on the y-axis and time (hr) on the x-axis, showing a scatter plot with a trend line.)

Explanation:

Step1: Identify slope (rate of change)

From the graph, when \( x = 0 \) (time = 0 hours), \( y = 37 \) (temperature). When \( x = 10 \) hours, \( y = 0 \) (assuming the end point). The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 37}{10 - 0}=-\frac{37}{10}=- 3.7\) (approx -3.7 per hour, representing temperature drop per hour).

Step2: Identify y - intercept

The y - intercept (\( b \)) is the value of \( y \) when \( x = 0 \). From the graph, at \( x = 0 \), \( y = 37 \). So \( b = 37 \), representing the initial temperature (when the cold front arrived, \( x = 0 \)).

Step3: Formulate linear equation

The slope - intercept form of a line is \( y=mx + b \). We found \( m=-3.7\) (or \( m =-\frac{37}{10}\)) and \( b = 37 \). So the equation is \( y=-3.7x + 37 \) (or \( y =-\frac{37}{10}x+37\)).

Answer:

For part c, the equation is \( y=-3.7x + 37 \) (or \( y =-\frac{37}{10}x + 37\))