Sure! Let's break this down and explain the answer step-by-step based on the analysis of the models proposed by Clara and Michael.
First, let's recall the context:
- We have the amounts raised in the first 9 months.
- We need to use Clara's and Michael's functions to predict the amounts for the next three months, i.e., Months 10, 11, and 12.
### Clara's Predictions:
Clara's function is given by:
[tex]\[ y = -3.14 x^2 + 44.7 x + 203.6 \][/tex]
To find the expected amount for Month 10:
[tex]\[ y_{10} = -3.14 (10)^2 + 44.7 \cdot 10 + 203.6 \approx 336.6 \][/tex]
For Month 11:
[tex]\[ y_{11} = -3.14 (11)^2 + 44.7 \cdot 11 + 203.6 \][/tex]
For Month 12:
[tex]\[ y_{12} = -3.14 (12)^2 + 44.7 \cdot 12 + 203.6 \][/tex]
Based on the function outcome for Month 10, Clara's prediction for the amount of money the club would expect to raise each month over the next three months is:
[tex]\[ \boxed{336.6} \][/tex]
### Michael's Predictions:
Michael's function is given by:
[tex]\[ y = 44.64 \sqrt{x+1} + 246.5 \][/tex]
To find the expected amount for Month 10:
[tex]\[ y_{10} = 44.64 \sqrt{10 + 1} + 246.5 \][/tex]
For Month 11:
[tex]\[ y_{11} = 44.64 \sqrt{11 + 1} + 246.5 \][/tex]
For Month 12:
[tex]\[ y_{12} = 44.64 \sqrt{12 + 1} + 246.5 \][/tex]
Summing Michael's predictions for the next three months:
[tex]\[ y_{10} + y_{11} + y_{12} \approx 1203.143 \][/tex]
Thus, Michael's prediction for the total amount of money the club would expect to raise over the next three months is:
[tex]\[ \boxed{1203.143} \][/tex]
### Summary:
- If the club uses Clara's function, it would expect to raise about 336.6 dollars each month.
- If the club uses Michael's function, it would expect to raise a total of approximately 1203.143 dollars over the next three months.
