Step 1

Answer for (a): The following table lists the reported number of cases of infants born in the United States with HIV in recent years because their mother was infected

\(\begin{array}{|c|c|}\hline \text{Year} & \text{Cases} \\ \hline 1995 & 295 \\ \hline 1997 & 166 \\ \hline 1999 & 109 \\ \hline 2001 & 115 \\ \hline 2003 & 94 \\ \hline 2005 & 107 \\ \hline 2007 & 79 \\ \hline \end{array}\)

Now letting \(\displaystyle{t}={0}\) corresponds to the year 1995 we get the following table.

\(\begin{array}{|c|c|}\hline \text{Year} & t & \text{Cases} \\ \hline 1995 & 0 & 295 \\ \hline 1997 & 2 & 166 \\ \hline 1999 & 4 & 109 \\ \hline 2001 & 6 & 115 \\ \hline 2003 & 8 & 94 \\ \hline 2005 & 10 & 107 \\ \hline 2007 & 12 & 79 \\ \hline \end{array}\)

Step 2

By using graphical calculator we have the following graph,

Step 3

Answer for (b) Using the regression feature on our calculator, a quadratic, a cubic, and an exponential function that models the above data are given as

\(\displaystyle{y}={2.3839}{t}^{{{2}}}\ -\ {42.5536}{t}\ +\ {269.2143}\)

\(\displaystyle{y}=\ -{0.4931}{t}^{{{3}}}\ +\ {1.2589}{t}^{{{2}}}\ -\ {81.998}{t}\ +\ {292.881}\)

\(\displaystyle{y}={213.7921}{\left({0.9149}\right)}^{{{t}}}\)

Answer for(c) Let us plot above three function and data point on the same window we observe that the exponential function \(\displaystyle{y}={213.7921}{\left({0.9149}\right)}^{{{t}}}\) is the best fit the data.

Step 4

Answer for (d) \(\displaystyle{t}={20}\) corresponds to the year 2015.

\(\displaystyle{y}{\left({20}\right)}={2.383}{\left({20}\right)}^{{{2}}}\ -\ {42.5536}{\left({20}\right)}\ +\ {269.2143}\)

\(\displaystyle={371.7023}\ \approx\ {372}\)

\(\displaystyle{y}{\left({20}\right)}=\ -{0.4931}{\left({20}\right)}^{{{3}}}\ +\ {11.2589}{\left({20}\right)}^{{{2}}}\ -\ {81.998}{\left({20}\right)}\ +\ {292.881}\)

\(\displaystyle=\ -{788.319}\ \approx\ -{788}\)

\(\displaystyle{y}{\left({20}\right)}={213.7921}{\left({0.9149}\right)}^{{{20}}}\)

\(\displaystyle={36.0964}\ \approx\ {36}\)

The only realistic value is given by the exponential function because the pattern of the data suggests that the number of cases decrease over time.