First, we will analyze the data given in the table:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Time, } x \text{ (hours)} & \text{Average Speed, } y \text{ (miles per hour)} \\
\hline
12 & 8 \\
\hline
16 & 6 \\
\hline
10 \frac{2}{3} & 9 \\
\hline
18 & 5 \frac{1}{3} \\
\hline
\end{array}
\][/tex]
To find a rational function [tex]\( y = \frac{1}{ax + b} \)[/tex] that best models the data, we will proceed with the following steps:
1. Convert Speeds to Reciprocal Values:
The rational function is in the form of [tex]\( y = \frac{1}{ax + b} \)[/tex], equivalent to rewriting the speeds to their reciprocals because we will linearize the function.
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Time, } x \text{ (hours)} & \text{Average Speed, } y \text{ (miles per hour)} & \text{Reciprocal Speed, } \frac{1}{y} \\
\hline
12 & 8 & \frac{1}{8} = 0.125 \\
\hline
16 & 6 & \frac{1}{6} \approx 0.1667 \\
\hline
10 \frac{2}{3} & 9 & \frac{1}{9} \approx 0.1111 \\
\hline
18 & 5 \frac{1}{3} & \frac{1}{\left( \frac{16}{3} \right)} = \frac{3}{16} \approx 0.1875 \\
\hline
\end{array}
\][/tex]
2. Fit a Linear Equation to Reciprocal Speeds:
We fit a linear function to the reciprocal speed data to find the coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex] in the linear form [tex]\( \frac{1}{y} = ax + b \)[/tex].
3. Determine the Coefficients:
By fitting a linear regression model to the reciprocal speed data, we find that:
[tex]\[
a \approx 0.0104, \quad b \approx 6.9724 \times 10^{-17}
\][/tex]
Therefore, the rational function that best models the data is:
[tex]\[
y = \frac{1}{0.0104x + 6.9724 \times 10^{-17}}
\][/tex]
To summarize, we transformed the original average speeds into their reciprocals, fitted a linear equation to these transformed values, and obtained the coefficients. These coefficients were then substituted back into the rational function form, yielding the final model:
[tex]\[ y = \frac{1}{0.0104x + 6.9724 \times 10^{-17}} \][/tex]
