To analyze the given data for Kimberly and Katrina, let's break down your questions one by one.
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The person doing the most jumping jacks per minute: To find out who did the most jumping jacks per minute, we can calculate the jumping jacks per minute for both individuals based on the data provided.
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Kimberly:
- At 3 minutes: \( \frac{28}{3} \approx 9.33 \) jacks/min
- At 8 minutes: \( \frac{58}{8} = 7.25 \) jacks/min
- At 12 minutes: \( \frac{82}{12} \approx 6.83 \) jacks/min
- At 16 minutes: \( \frac{106}{16} \approx 6.625 \) jacks/min
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Katrina:
- At 2 minutes: \( \frac{16}{2} = 8 \) jacks/min
- At 5 minutes: \( \frac{40}{5} = 8 \) jacks/min
- At 12 minutes: \( \frac{96}{12} = 8 \) jacks/min
- At 20 minutes: \( \frac{160}{20} = 8 \) jacks/min
The maximum jacks per minute for Kimberly is approximately 9.33, and for Katrina, it is consistently 8. Therefore, Kimberly is the person doing the most jumping jacks per minute.
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The person showing a proportional relationship between time and the number of jumping jacks: A proportional relationship means that as one variable changes, the other variable changes at a constant rate (i.e., the ratio remains constant).
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For Kimberly:
- The ratios are not constant since the jumping jacks per minute decrease as time increases, indicating a non-proportional relationship.
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For Katrina:
- The ratios are: \( \frac{16}{2} = 8 \), \( \frac{40}{5} = 8 \), \( \frac{96}{12} = 8 \), \( \frac{160}{20} = 8 \).
- As we can see, Katrina's jumping jacks per minute remains consistent at 8, indicating a proportional relationship.
Therefore, Katrina shows a proportional relationship between time and the number of jumping jacks.
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The person that did the most jumping jacks initially before the time started: This question pertains to the initial value of jumping jacks performed, often represented by the intercept of the relationship when time is 0. Based on the data provided, we don't have information right at time 0, but we can infer from the table.
- Kimberly: There is no data point at 0, so we can't determine her initial number.
- Katrina: Similar to Kimberly, there is no data point at 0.
Since there's no data for time 0 for both individuals, we can't conclusively determine who started with more jumping jacks. However, we know that both had all their counts starting from their respective first recorded times.
In summary:
- Kimberly does the most jumping jacks per minute.
- Katrina shows a proportional relationship between time and jumping jacks.
- We cannot identify who did the most jumping jacks initially before the time started based on the provided data.
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