To analyze the data provided, let's calculate the rate of jumping jacks per second for Kimberly and Katrina, examine their initial jumping jacks, and determine if there's a proportional relationship.
a. Jumping jacks per second
Kimberly:
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For the 3 seconds:
\( \frac{17 , \text{jacks}}{3 , \text{seconds}} \approx 5.67 , \text{jacks/second} \) -
For the 8 seconds:
\( \frac{37 , \text{jacks}}{8 , \text{seconds}} = 4.625 , \text{jacks/second} \) -
For the 12 seconds:
\( \frac{53 , \text{jacks}}{12 , \text{seconds}} \approx 4.42 , \text{jacks/second} \) -
For the 16 seconds:
\( \frac{69 , \text{jacks}}{16 , \text{seconds}} \approx 4.31 , \text{jacks/second} \)
Average jacks per second for Kimberly:
- Total jacks: 17 + 37 + 53 + 69 = 176
- Total seconds: 3 + 8 + 12 + 16 = 39
- Average: \( \frac{176 , \text{jacks}}{39 , \text{seconds}} \approx 4.51 , \text{jacks/second} \)
Katrina:
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For the 2 seconds:
\( \frac{10 , \text{jacks}}{2 , \text{seconds}} = 5 , \text{jacks/second} \) -
For the 5 seconds:
\( \frac{25 , \text{jacks}}{5 , \text{seconds}} = 5 , \text{jacks/second} \) -
For the 12 seconds:
\( \frac{60 , \text{jacks}}{12 , \text{seconds}} = 5 , \text{jacks/second} \) -
For the 20 seconds:
\( \frac{100 , \text{jacks}}{20 , \text{seconds}} = 5 , \text{jacks/second} \)
Average jacks per second for Katrina:
- Total jacks: 10 + 25 + 60 + 100 = 195
- Total seconds: 2 + 5 + 12 + 20 = 39
- Average: \( \frac{195 , \text{jacks}}{39 , \text{seconds}} = 5 , \text{jacks/second} \)
Conclusion for (a):
- Kimberly averaged around 4.51 jacks/second.
- Katrina had a steady rate of 5 jacks/second.
- Katrina is doing more jumping jacks per second.
b. Initial jumping jacks
To determine the initial jumping jacks before the timer started, we can look at the data provided:
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For Kimberly, all values start from time 3 seconds (17 jumping jacks), hence she started with 0 before she began.
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For Katrina, the first recorded value is at 2 seconds with 10 jumping jacks, so she had completed 10 jumping jacks before the timer started.
Conclusion for (b):
- Katrina had done more jumping jacks initially before the timer started. (10 vs. 0)
c. Proportional relationship
A proportional relationship means that the ratio of jumping jacks to time remains constant.
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For Kimberly, the ratio fluctuates. For example:
- \( 17 / 3 \approx 5.67 \)
- \( 37 / 8 = 4.625 \)
- \( 53 / 12 \approx 4.42 \)
- \( 69 / 16 \approx 4.31 \)
Because the ratios are not constant, this does not represent a proportional relationship.
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For Katrina, every time point yields a ratio of 5 \( \text{jacks/second} \):
- \( 10 / 2 = 5 \)
- \( 25 / 5 = 5 \)
- \( 60 / 12 = 5 \)
- \( 100 / 20 = 5 \)
Since Katrina has a constant ratio of jumping jacks to time, she shows a proportional relationship.
Conclusion for (c):
- Katrina shows a proportional relationship.