To determine the values of the coefficients for the quadratic model [tex]\( y = ax^2 + bx + c \)[/tex] that best fits the given data, we begin with the following data points:
[tex]\[
\begin{tabular}{|c|c|}
\hline
Time (Weeks) & Returns Filed (Millions) \\
\hline
2 & 39.75 \\
\hline
3 & 49.92 \\
\hline
5 & 67.72 \\
\hline
7 & 84.08 \\
\hline
10 & 119.43 \\
\hline
11 & 137.23 \\
\hline
13 & 141 \\
\hline
14 & 141.57 \\
\hline
\end{tabular}
\][/tex]
Fitting these data points to a quadratic function of the form [tex]\( y = ax^2 + bx + c \)[/tex], we obtain the coefficients for [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].
The specific values for these coefficients are:
- The value of [tex]\( a \)[/tex] is [tex]\( -0.24498881831119262 \)[/tex]
- The value of [tex]\( b \)[/tex] is [tex]\( 13.09543454869118 \)[/tex]
- The value of [tex]\( c \)[/tex] is [tex]\( 11.796778632313286 \)[/tex]
Thus, the sentences can be completed as follows:
The value of [tex]\( a \)[/tex] is [tex]\(-0.24498881831119262\)[/tex], the value of [tex]\( b \)[/tex] is [tex]\(13.09543454869118\)[/tex], and the value of [tex]\( c \)[/tex] is [tex]\(11.796778632313286\)[/tex].
