To determine which rational function best models the data in the table, we need to compare the fit of two potential models. These models are:
1. [tex]\( y = \frac{x}{96} \)[/tex]
2. [tex]\( y = \frac{2x}{3} \)[/tex]
Given the table of data points:
[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]
Let's check how well each model fits these data points.
### Model [tex]\( y = \frac{x}{96} \)[/tex]
1. For [tex]\( x = 12 \)[/tex]:
[tex]\[
y = \frac{12}{96} = 0.125
\][/tex]
Difference: [tex]\( |8 - 0.125| = 7.875 \)[/tex]
2. For [tex]\( x = 16 \)[/tex]:
[tex]\[
y = \frac{16}{96} \approx 0.1667
\][/tex]
Difference: [tex]\( |6 - 0.1667| \approx 5.8333 \)[/tex]
3. For [tex]\( x = 10.6667 \)[/tex]:
[tex]\[
y = \frac{10.6667}{96} \approx 0.1111
\][/tex]
Difference: [tex]\( |9 - 0.1111| \approx 8.8889 \)[/tex]
4. For [tex]\( x = 18 \)[/tex]:
[tex]\[
y = \frac{18}{96} \approx 0.1875
\][/tex]
Difference: [tex]\( |5.3333 - 0.1875| \approx 5.1458 \)[/tex]
### Model [tex]\( y = \frac{2x}{3} \)[/tex]
1. For [tex]\( x = 12 \)[/tex]:
[tex]\[
y = \frac{2 \times 12}{3} = \frac{24}{3} = 8
\][/tex]
Difference: [tex]\( |8 - 8| = 0 \)[/tex]
2. For [tex]\( x = 16 \)[/tex]:
[tex]\[
y = \frac{2 \times 16}{3} \approx 10.6667
\][/tex]
Difference: [tex]\( |6 - 10.6667| \approx 4.6667 \)[/tex]
3. For [tex]\( x = 10.6667 \)[/tex]:
[tex]\[
y = \frac{2 \times 10.6667}{3} \approx 7.1111
\][/tex]
Difference: [tex]\( |9 - 7.1111| \approx 1.8889 \)[/tex]
4. For [tex]\( x = 18 \)[/tex]:
[tex]\[
y = \frac{2 \times 18}{3} = 12
\][/tex]
Difference: [tex]\( |5.3333 - 12| \approx 6.6667 \)[/tex]
Now, let's compare the differences (errors) for both models:
- Errors for [tex]\( y = \frac{x}{96} \)[/tex]: [tex]\([7.875, 5.8333, 8.8889, 5.1458]\)[/tex]
- Errors for [tex]\( y = \frac{2x}{3} \)[/tex]: [tex]\([0, 4.6667, 1.8889, 6.6667]\)[/tex]
By comparing the magnitude of the errors, we see that the errors for [tex]\( y = \frac{2x}{3} \)[/tex] are generally smaller than those for [tex]\( y = \frac{x}{96} \)[/tex]. Therefore, the model [tex]\( y = \frac{2x}{3} \)[/tex] best fits the given data points in the table.
