To determine which equation best represents the data shown in the table, we need to evaluate the fit of each given equation to the data points:
[tex]\[
\begin{array}{|c|c|}
\hline \multicolumn{2}{|c|}{\text{Stopping Distance of a Car vs. Speed}} \\
\hline \text{Stopping Distance, } d \text{ (in feet)} & \text{Speed, } s \text{ (in miles per hour)} \\
\hline 20 & 17.89 \\
\hline 40 & 25.30 \\
\hline 60 & 30.98 \\
\hline 80 & 35.78 \\
\hline 100 & 40.00 \\
\hline
\end{array}
\][/tex]
We have four potential models to test:
1. [tex]\( s = 4 \sqrt{d} \)[/tex]
2. [tex]\( s = 0.37d + 20 \)[/tex]
3. [tex]\( s = 13.7 \log(d) \)[/tex]
4. [tex]\( s = \frac{1}{25} d^2 \)[/tex]
Using the given data, we examine which of these equations provides the best fit to the data points. This involves calculating the R-squared value for each model, which measures how well the model's predictions match the actual data.
Equations:
1. [tex]\( s = 4 \sqrt{d} \)[/tex]
2. [tex]\( s = 0.37d + 20 \)[/tex]
3. [tex]\( s = 13.7 \log(d) \)[/tex]
4. [tex]\( s = \frac{1}{25} d^2 \)[/tex]
Steps:
1. Calculate the predicted values of speed ([tex]\(s\)[/tex]) for each stopping distance ([tex]\(d\)[/tex]) using the equations.
2. Compute the residuals (the differences between the actual speeds and the predicted speeds).
3. Calculate the R-squared value for each model to evaluate the goodness of fit.
Result:
Upon evaluating the R-squared values from the four models, the equation [tex]\( s = 4 \sqrt{d} \)[/tex] is found to have the highest R-squared value, indicating that it best represents the given data.
Hence, the equation that best represents the relationship between stopping distance and speed is:
[tex]\[ s = 4 \sqrt{d} \][/tex]
