A local race for charity has taken place since 1993. Using...

A local race for charity has taken place since 1993. Using the actual speeds of the winners from 1993 through 1998, mathematicians obtained the function $f(x) = 0.17x + 4$, in which $x$ represents the number of years after 1993 and $f(x)$ represents the winning speed, in miles per hour. In what year is the winning speed predicted to be 6.04 mph?

The following table lists the qualifying speeds, in miles per hour, of the Indianapolis 500 car race winners for selected years from 1931 to $2005 .$ (Source: www. indy500.com) \begin{array}{|c|c|} \hline\text { Year } & \text { Qualifying Speed } \\\hline 1931 & (\mathrm{mph}) \\1941 & 107 \\1951 & 121 \\1961 & 135 \\1971 & 145 \\1981 & 174 \\1991 & 200 \\2005 & 224 \\228 & 228 \\\hline\end{array} (a) Fxplain why a logistic function would fit this data well. (b) Make a scatter plot of the data and find the logistic function of the form $f(x)=\frac{\epsilon}{1+a e^{-x}}$ that best fits the data, Let $x$ be the number of years since 1931 (c) What does $c$ signify in your model? (d) Using your model, what is the projected qualifying speed for the winner in $2008 ?$

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