Question

17. The tables below show the number of jumping jacks completed after a given period of time in seconds.Kimberly:                                                                       Time (seconds) Jumping Jacks3 178 3712 5316 69 Katrina: Time (seconds) Jumping Jacks2 105 2512 6020 100(3 points)a. Which person is doing more jumping jacks per second? b. Which person had done more jumping jacks initially before the timer started? c. Which person shows a proportional relationship?

Answer 1

To analyze the data for Kimberly and Katrina, we can complete each part of the question step by step:

a. Who is doing more jumping jacks per second?

To find the jumping jacks per second for each person, we'll divide the number of jumping jacks by the time in seconds.

Kimberly:

• 3 seconds: \( \frac{17}{3} \approx 5.67 \) JJs/second
• 8 seconds: \( \frac{37}{8} = 4.625 \) JJs/second
• 12 seconds: \( \frac{53}{12} \approx 4.42 \) JJs/second
• 16 seconds: \( \frac{69}{16} \approx 4.31 \) JJs/second

Average for Kimberly: \( \frac{17 + 37 + 53 + 69}{3 + 8 + 12 + 16} = \frac{176}{39} \approx 4.51 \) JJs/second

Katrina:

• 2 seconds: \( \frac{10}{2} = 5 \) JJs/second
• 5 seconds: \( \frac{25}{5} = 5 \) JJs/second
• 12 seconds: \( \frac{60}{12} = 5 \) JJs/second
• 20 seconds: \( \frac{100}{20} = 5 \) JJs/second

Average for Katrina: Since all are consistent, Katrina has \( 5 \) JJs/second.

Conclusion for a: Katrina is doing more jumping jacks per second, averaging 5 JJs/second compared to Kimberly's average of approximately 4.51 JJs/second.

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b. Who has done more jumping jacks initially before the timer started?

From the data provided, we can look at the values associated with the starting point.

• It appears that there’s no direct mention of jumping jacks completed before the timer started. However, since both tables start with a time of 2 seconds for Katrina and 3 seconds for Kimberly, we can deduce that:

• Katrina: At 2 seconds, she completed 10 jumping jacks.

• Kimberly: At 3 seconds, she completed 17 jumping jacks.

From this indirect insight:

Conclusion for b: Kimberly had done more jumping jacks before the timer started (i.e. at 3 seconds had 17 jumping jacks).

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c. Which person shows a proportional relationship?

To determine if the relationship is proportional, we need to check if the ratio of jumping jacks to time remains constant (i.e., \( k = \frac{y}{x} \) is consistent across all data points).

For Kimberly:

1. 3 seconds: \( \frac{17}{3} \)
2. 8 seconds: \( \frac{37}{8} \)
3. 12 seconds: \( \frac{53}{12} \)
4. 16 seconds: \( \frac{69}{16} \)

The ratios above are not constant.

For Katrina:

1. 2 seconds: \( \frac{10}{2} = 5 \)
2. 5 seconds: \( \frac{25}{5} = 5 \)
3. 12 seconds: \( \frac{60}{12} = 5 \)
4. 20 seconds: \( \frac{100}{20} = 5 \)

The ratio of Katrina’s jumping jacks to time is consistent across all intervals.

Conclusion for c: Katrina shows a proportional relationship, as her jumping jacks per second remains constant at 5 JJs/second throughout her recorded times.