\begin{tabular}{l} The equation for the line of best fit is $$ y=20 x-600 $$. \ In this context, what is the meaning of 20 ? \ An increase of $$ 20^{\circ} \mathrm{F} $$ is associated with 1 more smoothie being sold. \ An increase of $$ 1^{\circ} \mathrm{F} $$ is associated with 20 more smoothies being sold. \ Weekly sales should never drop below 20 smoothies. \ Weekly sales should be 20 smoothies at a temperature of $$ 0^{\circ} \mathrm{F} $$. \ Wherage temperature ( \ Wh) \ \hline\end{tabular}

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UpStudy AI · Accepted Answer

To interpret the meaning of the slope (20) in the equation of the line of best fit $$ y = 20x - 600 $$, we need to analyze the components of the equation.

1. **Understanding the Equation**:
   - The equation is in the form $$ y = mx + b $$, where:
     - $$ m $$ is the slope (20 in this case).
     - $$ b $$ is the y-intercept (-600 in this case).
     - $$ x $$ represents the independent variable (temperature in degrees Fahrenheit).
     - $$ y $$ represents the dependent variable (number of smoothies sold).

2. **Interpreting the Slope (20)**:
   - The slope of the line (20) indicates the rate of change of the dependent variable (smoothies sold) with respect to the independent variable (temperature).
   - Specifically, a slope of 20 means that for every 1 degree Fahrenheit increase in temperature, the number of smoothies sold increases by 20.

3. **Evaluating the Given Statements**:
   - **"An increase of $$ 20^{\circ} \mathrm{F} $$ is associated with 1 more smoothie being sold."**: This is incorrect. An increase of $$ 20^{\circ} \mathrm{F} $$ would lead to an increase of $$ 20 \times 20 = 400 $$ smoothies sold.
   - **"An increase of $$ 1^{\circ} \mathrm{F} $$ is associated with 20 more smoothies being sold."**: This is correct. This directly reflects the meaning of the slope.
   - **"Weekly sales should never drop below 20 smoothies."**: This statement is not directly related to the slope and requires additional context.
   - **"Weekly sales should be 20 smoothies at a temperature of $$ 0^{\circ} \mathrm{F} $$."**: To check this, we can substitute $$ x = 0 $$ into the equation:
     $$
     y = 20(0) - 600 = -600
     $$
     This indicates that at $$ 0^{\circ} \mathrm{F} $$, the model predicts -600 smoothies, which is not feasible. Therefore, this statement is incorrect.

4. **Conclusion**:
   - The correct interpretation of the slope (20) is that an increase of $$ 1^{\circ} \mathrm{F} $$ is associated with 20 more smoothies being sold.
An increase of $$ 1^{\circ} \mathrm{F} $$ is associated with 20 more smoothies being sold.

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