Let's analyze and solve this problem step-by-step.
1. Understanding the Problem:
We are given a quadratic regression equation that models the rainfall data for a city across different months:
[tex]\[
y = -0.77x^2 + 6.06x - 5.9
\][/tex]
where [tex]\( x \)[/tex] represents the month, and [tex]\( y \)[/tex] represents the rainfall in centimeters.
2. Determine Predictions Using the Model:
We are asked to predict the rainfall for month 11 using this model. Substituting [tex]\( x = 11 \)[/tex] into the regression equation:
[tex]\[
y = -0.77(11)^2 + 6.06(11) - 5.9
\][/tex]
We are given that the result of this calculation is:
[tex]\[
y \approx -32.4 \, \text{centimeters}
\][/tex]
3. Review the Result:
The predicted rainfall for month 11 is approximately -32.4 centimeters.
4. Interpret the Prediction:
Now, let's determine if this prediction makes sense by examining the possible answer choices:
- A. No, because you can't have a negative amount of rainfall.
- B. Yes, because it is the result of substituting [tex]\( x = 11 \)[/tex].
- C. Yes, because the rainfall is declining.
- D. No, because you can't measure rainfall in centimeters.
Evaluate each option:
- Option B: The result is indeed from substituting [tex]\( x = 11 \)[/tex] into the equation, but just because we followed the mathematical process doesn't validate the physical feasibility of negative rainfall.
- Option C: While the model shows a declining trend, it doesn't support the idea that the rainfall can be negative.
- Option D: Rainfall can be measured in centimeters, so this is incorrect.
- Option A: Negative rainfall is not physically possible since rainfall represents an amount of precipitation, which cannot be less than zero.
5. Conclusion:
Considering the physical reality of rainfall, which cannot be negative, the only logical answer is:
- A. No, because you can't have a negative amount of rainfall.
Therefore, the prediction of -32.4 centimeters for month 11 does not make sense because it results in a negative amount of rainfall, which is not possible.
