Select the correct answer from each drop-down menu.

A school club will be competing at a state championship and has been working to raise money for the club's travel expenses. The table shows the amount of money raised each month over a nine-month period beginning in August.

\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}
\hline Month & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline Amount (\$) & 250 & 275 & 325 & 300 & 350 & 375 & 375 & 350 & 350 \\
\hline
\end{tabular}

Based on the data in the table, Clara and Michael each use their own function to determine the amount of money the club should expect to raise in the next three months.

Clara's function: [tex]$y=-3.14 x^2+44.7 x+203.6$[/tex]
Michael's function: [tex]$y=44.64 \sqrt{x+1}+246.5$[/tex]

How does using the different models affect the amount of money the club would expect to raise in the next three months?

If the club uses Clara's function, it would expect the amount of money to [tex]$\square$[/tex] each month.
If the club uses Michael's function, it would expect the amount of money to [tex]$\square$[/tex] each month.



Answer :

To determine the amount of money the club should expect to raise in the next three months, Clara and Michael used their respective functions to make predictions. Here is the step-by-step process to find their predictions:

1. Clara's function is given by:
[tex]\[ y = -3.14x^2 + 44.7x + 203.6 \][/tex]
We need to evaluate this function for the next three months:
- For month 10 (\(x = 10\)):
[tex]\[ y_{10} = -3.14(10)^2 + 44.7(10) + 203.6 = -314 + 447 + 203.6 = 336.6 \][/tex]
- For month 11 (\(x = 11\)):
[tex]\[ y_{11} = -3.14(11)^2 + 44.7(11) + 203.6 = -379.54 + 491.7 + 203.6 = 315.76 \][/tex]
- For month 12 (\(x = 12\)):
[tex]\[ y_{12} = -3.14(12)^2 + 44.7(12) + 203.6 = -452.16 + 536.4 + 203.6 = 287.44 \][/tex]
To find the expected monthly average:
[tex]\[ \text{Clara's expected average} = \frac{336.6 + 315.76 + 287.44}{3} = 313.27 \][/tex]

2. Michael's function is given by:
[tex]\[ y = 44.64\sqrt{x + 1} + 246.5 \][/tex]
We need to evaluate this function for the next three months:
- For month 10 (\(x = 10\)):
[tex]\[ y_{10} = 44.64\sqrt{10 + 1} + 246.5 = 44.64\sqrt{11} + 246.5 \approx 44.64 \times 3.3166 + 246.5 \approx 296.83 + 246.5 = 543.33 \][/tex]
- For month 11 (\(x = 11\)):
[tex]\[ y_{11} = 44.64\sqrt{11 + 1} + 246.5 = 44.64\sqrt{12} + 246.5 \approx 44.64 \times 3.4641 + 246.5 \approx 154.65 + 246.5 = 542.63 \][/tex]
- For month 12 (\(x = 12\)):
[tex]\[ y_{12} = 44.64\sqrt{12 + 1} + 246.5 = 44.64\sqrt{13} + 246.5 \approx 44.64 \times 3.6056 + 246.5 \approx 161.02 + 246.5 = 540.83 \][/tex]
To find the expected monthly average:
[tex]\[ \text{Michael's expected average} = \frac{543.33 + 542.63 + 540.83}{3} = 542.93 \][/tex]

Therefore, based on these calculations, if the club uses Clara's function, it would expect the amount of money to be around [tex]$313.27 each month. If the club uses Michael's function, it would expect the amount of money to be around $[/tex]401.05 each month.