It takes Julian 1/2 hour to walk 2 miles. He decides to start walking in his spare time, but because he wants to make sure he has enough time to go a desired distance. He created the data table at which ratio does the constant proportionality appear.
To determine at which ratio the constant proportionality appears, we need to analyze the relationship between time and distance that Julian has observed. According to the problem, Julian takes 1/2 hour to walk 2 miles. We can create a table to represent this relationship:
| Time (hours) | Distance (miles) |
|--------------|-----------------|
| 1/2 | 2 |
| 1 | ? |
| 2 | ? |
| 3 | ? |
| 4 | ? |
| ... | ... |
We want to find a pattern in the data table. Let's see how distance changes as time increases:
- From the given information, Julian takes 1/2 hour to walk 2 miles. This means that his speed or rate is 2 miles per 1/2 hour, which simplifies to 4 miles per hour.
- If Julian continues to walk at the same speed, after 1 hour he will have covered 4 miles, and after 2 hours he will have covered 8 miles.
- Therefore, we can fill in the table as follows:
| Time (hours) | Distance (miles) |
|--------------|-----------------|
| 1/2 | 2 |
| 1 | 4 |
| 2 | 8 |
| 3 | 12 |
| 4 | 16 |
| ... | ... |
By analyzing the data table, we observe that as the time increases by a constant factor, the distance also increases by a constant factor. For example, when the time doubles from 1/2 hour to 1 hour, the distance doubles from 2 miles to 4 miles. This indicates a constant ratio of 2:1 between time and distance. Therefore, the constant proportionality appears at a ratio of 2:1 between time and distance.