To determine if the table shows a proportional relationship, we need to check whether the ratio of pay to time is consistent across all pairs of values given.
The table provides the following pairs of time in hours [tex]\( t \)[/tex] and pay in dollars [tex]\( P \)[/tex]:
[tex]\[
\begin{array}{c|c|c|c|c|c}
\text{Time (hours)} & 0 & 8 & 16 & 24 & 32 \\
\hline
\text{Pay (dollars)} & 0 & 96 & 192 & 312 & 416 \\
\end{array}
\][/tex]
We'll calculate the ratio [tex]\( \frac{P}{t} \)[/tex] for each nonzero time:
1. For [tex]\( t = 8 \)[/tex] hours:
[tex]\[
\frac{96}{8} = 12 \text{ dollars per hour}
\][/tex]
2. For [tex]\( t = 16 \)[/tex] hours:
[tex]\[
\frac{192}{16} = 12 \text{ dollars per hour}
\][/tex]
3. For [tex]\( t = 24 \)[/tex] hours:
[tex]\[
\frac{312}{24} \approx 13 \text{ dollars per hour}
\][/tex]
4. For [tex]\( t = 32 \)[/tex] hours:
[tex]\[
\frac{416}{32} = 13 \text{ dollars per hour}
\][/tex]
We observe the ratios calculated:
- [tex]\( \frac{96}{8} = 12 \)[/tex]
- [tex]\( \frac{192}{16} = 12 \)[/tex]
- [tex]\( \frac{312}{24} = 13 \)[/tex]
- [tex]\( \frac{416}{32} = 13 \)[/tex]
Since these ratios are not all the same (i.e., some are 12 and others 13 dollars per hour), this indicates that the ratios of pay to time are not equivalent across all periods.
Therefore, the correct answer is:
No, it is not proportional because [tex]\( \frac{96}{8} \neq \frac{312}{24} \)[/tex].
