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The adjacent table gives the number of worldwide Internet users, in millions.

Complete parts (a) through (c).

\begin{tabular}{|c|c|c|c|}
\hline Year & Users (millions) & Year & Users (millions) \\
\hline 1995 & 10 & 2002 & 523 \\
\hline 1996 & 38 & 2003 & 713 \\
\hline 1997 & 75 & 2004 & 820 \\
\hline 1998 & 183 & 2005 & 1043 \\
\hline 1999 & 295 & 2006 & 1099 \\
\hline 2000 & 341 & 2007 & 1216 \\
\hline 2001 & 532 & & \\
\hline
\end{tabular}

(a) Find the cubic function that is the best fit for the data, with [tex]$y$[/tex] equal to the number of millions of users and [tex]$x$[/tex] equal to the number of years from 1990.

[tex]
y=
[/tex]
[tex]\square[/tex] (Use integers or decimals for any numbers in the expression. Round to three decimal places as needed.)

Sagot :

GinnyAnsweridn
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]
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