To find the cubic function that best fits the given data, we'll use [tex]\( y \)[/tex] to represent the number of millions of users and [tex]\( x \)[/tex] to represent the number of years from 1990.
The given years and corresponding number of users are as follows:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Year} & \text{Users (millions)} \\
\hline 1995 & 10 \\
1996 & 38 \\
1997 & 75 \\
1998 & 183 \\
1999 & 295 \\
2000 & 341 \\
2001 & 532 \\
2002 & 523 \\
2003 & 713 \\
2004 & 820 \\
2005 & 1043 \\
2006 & 1099 \\
2007 & 1216 \\
\hline
\end{array}
\][/tex]
To translate the years into the number of years from 1990, we subtract 1990 from each year:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Year} & x = \text{Years from 1990} \\
\hline 1995 & 5 \\
1996 & 6 \\
1997 & 7 \\
1998 & 8 \\
1999 & 9 \\
2000 & 10 \\
2001 & 11 \\
2002 & 12 \\
2003 & 13 \\
2004 & 14 \\
2005 & 15 \\
2006 & 16 \\
2007 & 17 \\
\hline
\end{array}
\][/tex]
We now need to find the coefficients of the cubic polynomial [tex]\( y = ax^3 + bx^2 + cx + d \)[/tex] that best fits this data. The coefficients can be determined through a method called polynomial regression.
The resulting cubic function that provides the best fit for our data is:
[tex]\[ y = -0.376x^3 + 16.389x^2 - 108.220x + 182.728 \][/tex]
So, the best fit cubic function rounded to three decimal places is:
[tex]\[ y = -0.376x^3 + 16.389x^2 - 108.220x + 182.728 \][/tex]
