To analyze the jumping jacks completed by Kimberly and Katrina, we can calculate the rate of jumping jacks per second for each person and assess their initial counts when the timer starts.
a. Which person is doing more jumping jacks per second?
Kimberly's Jumping Jacks per Second:
- At 3 seconds: \( \frac{17}{3} \approx 5.67 \)
- At 8 seconds: \( \frac{37}{8} = 4.625 \)
- At 12 seconds: \( \frac{53}{12} \approx 4.42 \)
- At 16 seconds: \( \frac{69}{16} \approx 4.31 \)
Average for Kimberly:
To find the overall average, we sum the total jumping jacks and divide by the total time:
\[
\text{Total JJs} = 17 + 37 + 53 + 69 = 176
\]
\[
\text{Total time} = 3 + 8 + 12 + 16 = 39
\]
\[
\text{Average JJs per second for Kimberly} = \frac{176}{39} \approx 4.51
\]
Katrina's Jumping Jacks per Second:
- At 2 seconds: \( \frac{10}{2} = 5 \)
- At 5 seconds: \( \frac{25}{5} = 5 \)
- At 12 seconds: \( \frac{60}{12} = 5 \)
- At 20 seconds: \( \frac{100}{20} = 5 \)
Average for Katrina:
Katrina has a constant rate of:
\[
\text{Average JJs per second for Katrina} = 5
\]
Conclusion for part a:
Katrina is doing more jumping jacks per second (5) compared to Kimberly (approximately 4.51).
b. Which person had done more jumping jacks initially before the timer started?
For both individuals, we can look at their data:
- Kimberly has done 17 jumping jacks at 3 seconds.
- Katrina has done 10 jumping jacks at 2 seconds.
Since neither person had any jumping jacks stated to have happened before the timer (time zero), we consider the starting point. However, we can see at their first recorded time, Kimberly had done more jumping jacks (17) at time 3 seconds compared to Katrina (10) at time 2 seconds.
Conclusion for part b:
Kimberly had completed more jumping jacks initially prior to the timer starting.
c. Which person shows a proportional relationship?
A proportional relationship means that the ratio of jumping jacks to time is constant.
Kimberly:
- Ratios are not constant:
- Time = 3s, JJs = 17 → \( \frac{17}{3} \)
- Time = 8s, JJs = 37 → \( \frac{37}{8} \)
- Others also show different ratios.
Katrina:
- All ratios are equal (5):
- Time = 2s, JJs = 10 → \( \frac{10}{2} = 5 \)
- Time = 5s, JJs = 25 → \( \frac{25}{5} = 5 \)
- Time = 12s, JJs = 60 → \( \frac{60}{12} = 5 \)
- Time = 20s, JJs = 100 → \( \frac{100}{20} = 5 \)
Conclusion for part c:
Katrina shows a proportional relationship.
Final Answers:
a. Katrina is doing more jumping jacks per second.
b. Kimberly had done more jumping jacks initially.
c. Katrina shows a proportional relationship.
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