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.
