To determine which runners have speeds that are in a proportional relationship, we need to calculate the speed (in miles per second) for each runner. The formula for speed is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
We'll calculate the speed for each runner using the provided time (in seconds) and distance (in miles):
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Liam: \[ \text{Speed} = \frac{0.5}{306.6} \approx 0.00163 \text{ miles/second} \]
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Taylor: \[ \text{Speed} = \frac{1.2}{756.36} \approx 0.00159 \text{ miles/second} \]
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Sarah: \[ \text{Speed} = \frac{0.7}{504.35} \approx 0.00139 \text{ miles/second} \]
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Ashley: \[ \text{Speed} = \frac{0.75}{459.9} \approx 0.00163 \text{ miles/second} \]
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Connor: \[ \text{Speed} = \frac{1}{600.5} \approx 0.00167 \text{ miles/second} \]
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Nathan: \[ \text{Speed} = \frac{1.5}{942.75} \approx 0.00159 \text{ miles/second} \]
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Juan: \[ \text{Speed} = \frac{1.35}{827.82} \approx 0.00163 \text{ miles/second} \]
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Katie: \[ \text{Speed} = \frac{0.6}{429.48} \approx 0.00140 \text{ miles/second} \]
Now we can summarize the speeds:
- Liam: \( \approx 0.00163 \)
- Taylor: \( \approx 0.00159 \)
- Sarah: \( \approx 0.00139 \)
- Ashley: \( \approx 0.00163 \)
- Connor: \( \approx 0.00167 \)
- Nathan: \( \approx 0.00159 \)
- Juan: \( \approx 0.00163 \)
- Katie: \( \approx 0.00140 \)
To find runners in a proportional relationship, we look for runners where the ratio of distance to time remains constant.
From the calculations, we see:
- Liam, Ashley, and Juan all have a speed of approximately \(0.00163\).
- Taylor and Nathan both have a speed of approximately \(0.00159\).
Thus, the runners whose speeds are in a proportional relationship as they have the same ratio of distance to time are:
- Liam
- Ashley
- Juan
And
- Taylor
- Nathan
These pairs are in a proportional relationship based on their average speeds.