To answer the questions, we will analyze the data provided.
Data Summary:
User 1:
- 3 seconds: 17 jumping jacks
- 8 seconds: 37 jumping jacks
- 12 seconds: 53 jumping jacks
- 16 seconds: 69 jumping jacks
User 2 (Katrina):
- 2 seconds: 10 jumping jacks
- 5 seconds: 25 jumping jacks
- 12 seconds: 60 jumping jacks
- 20 seconds: 100 jumping jacks
a. Which person is doing more jumping jacks per second?
To find out who is doing more jumping jacks per second, we calculate the rate of jumping jacks per second for both persons.
User 1:
- At 3 seconds: \( \frac{17}{3} \approx 5.67 \) Jacks/second
- At 8 seconds: \( \frac{37}{8} \approx 4.625 \) Jacks/second
- At 12 seconds: \( \frac{53}{12} \approx 4.42 \) Jacks/second
- At 16 seconds: \( \frac{69}{16} \approx 4.31 \) Jacks/second
Average rate for User 1 is \( \frac{17 + 37 + 53 + 69}{3 + 8 + 12 + 16} = \frac{176}{39} \approx 4.51 \) Jacks/second
User 2 (Katrina):
- At 2 seconds: \( \frac{10}{2} = 5 \) Jacks/second
- At 5 seconds: \( \frac{25}{5} = 5 \) Jacks/second
- At 12 seconds: \( \frac{60}{12} = 5 \) Jacks/second
- At 20 seconds: \( \frac{100}{20} = 5 \) Jacks/second
Average rate for Katrina is also \( 5 \) Jacks/second.
Conclusion for part a: Katrina is doing more jumping jacks per second (5 Jacks/second vs. approximately 4.51 Jacks/second for User 1).
b. Which person had done more jumping jacks initially before the timer started?
We can find the initial number of jumping jacks for both persons by looking at available data points. Since there is no explicit mention of prior jumping jacks for both clients before their timers started, we can assume they started from 0.
Conclusion for part b:
Neither person had done any jumping jacks before the timer started (assuming they both started at 0).
c. Which person shows a proportional relationship?
A proportional relationship means that the rate of jumping jacks per second is constant. Let's check the data:
User 1:
- The rates of jumping jacks per second are not constant (decreasing over time).
Katrina:
- The rates of jumping jacks per second are constant at 5 Jacks/second.
Conclusion for part c: Katrina shows a proportional relationship since she has a constant rate of 5 jumping jacks per second for all intervals measured.
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