To find the ratio that represents the constant of proportionality in the data table provided, we need to analyze the relationship between the hours spent walking and the miles walked.
The values from the table are:
- (14 hours, 1 mile)
- (12 hours, 2 miles)
- (34 hours, 3 miles)
- (1 hour, 4 miles)
To find the ratio representing the constant of proportionality (which is typically the ratio of miles to hours or hours to miles), we can look for the relationship between miles and hours.
For each data point, we can compute the miles per hour:
- For (14, 1): \( \frac{1}{14} \) miles per hour.
- For (12, 2): \( \frac{2}{12} = \frac{1}{6} \) miles per hour.
- For (34, 3): \( \frac{3}{34} \) miles per hour.
- For (1, 4): \( \frac{4}{1} = 4 \) miles per hour.
The constant of proportionality in this case is not consistent across the data as the ratio changes depending on the time and distance.
However, looking through the provided responses:
- (4, 1) corresponds to 4 miles in 1 hour, which suggests a specific instance from the data.
- The ratios (14, 1), (1, 4), and \(\left(\frac{1}{4}, 1\right)\) imply different segments of the relationship.
The correct choice that resembles the constant of proportionality in a notable way is the ratio that can be interpreted as consistent with our goal of understanding how many miles per hour Julian walks when he efficiently uses his time:
- Thus, \((1, 4)\) suggests he walks 4 miles in 1 hour, which is a valid representation.
So the answer to your question is: (1, 4).