To interpret the line of best fit equation [tex]\( y = -1.5x + 173 \)[/tex] for the given data, let's break down the components of this linear equation and understand what they represent in this context.
1. Understanding the equation [tex]\( y = -1.5x + 173 \)[/tex]:
- Here, [tex]\( y \)[/tex] represents the temperature in degrees Fahrenheit.
- [tex]\( x \)[/tex] represents the time in minutes that a member can tolerate the heat in the sauna.
- The equation forms a straight line on a graph.
2. Y-intercept:
- The y-intercept is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
- In the given equation, the y-intercept is [tex]\( 173 \)[/tex].
- This means that when the time ([tex]\( x \)[/tex]) is 0 minutes, the temperature ([tex]\( y \)[/tex]) is 173 degrees Fahrenheit.
3. Slope:
- The slope of the line is [tex]\(-1.5\)[/tex].
- The slope indicates the rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex].
- A slope of [tex]\(-1.5\)[/tex] means that for every increase of 1 minute in the time ([tex]\( x \)[/tex]), the temperature ([tex]\( y \)[/tex]) decreases by 1.5 degrees Fahrenheit.
4. Interpreting the given options:
- Option 1: "The member can tolerate a temperature of 173 degrees Fahrenheit for 0 minutes."
- This interpretation is correct based on the y-intercept. When [tex]\( x = 0 \)[/tex], [tex]\( y = 173 \)[/tex].
- Option 2: "The amount of time the member can tolerate the heat in a sauna is 173 minutes."
- This interpretation does not align with the equation. The y-intercept of 173 is a temperature, not a time duration.
- Option 3: "The time increased 1.5 minutes for every degree Fahrenheit the temperature increased."
- This interpretation is incorrect. The slope of [tex]\(-1.5\)[/tex] suggests that temperature decreases as time increases, not the other way around.
- Option 4: "The time decreased 1.5 minutes for every degree Fahrenheit the temperature decreased."
- This interpretation is also incorrect. The correct understanding is that the temperature decreases by 1.5 degrees Fahrenheit for every minute of time that increases.
From the analysis above, the correct interpretation of the line of best fit [tex]\( y = -1.5x + 173 \)[/tex] is:
"The member can tolerate a temperature of 173 degrees Fahrenheit for 0 minutes."
