The tables below show the number of jumping jacks completed after a given period of time in seconds.
Kimberly:
Time (seconds)
• 3
• 8
• 12
• 16
Jumping Jacks
• 17
• 37
• 53
• 69
katrina:
Time (seconds)
• 2
• 5
• 12
• 20
Jumping Jacks
• 10
• 25
• 60
• 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? _
a. To determine which person is doing more jumping jacks per second, we can calculate the jumping jacks per second for each person.
For Kimberly:
- between 3-8 seconds: (37-17) / (8-3) = 4 jumping jacks per second
- between 8-12 seconds: (53-37) / (12-8) = 4 jumping jacks per second
- between 12-16 seconds: (69-53) / (16-12) = 4 jumping jacks per second
For Katrina:
- between 2-5 seconds: (25-10) / (5-2) = 5 jumping jacks per second
- between 5-12 seconds: (60-25) / (12-5) = 7.86 jumping jacks per second
- between 12-20 seconds: (100-60) / (20-12) = 8.33 jumping jacks per second
Therefore, Katrina is doing more jumping jacks per second.
b. To determine who had done more jumping jacks initially before the timer started, we look at the data at 0 seconds for each person. However, the information at 0 seconds is not provided in the tables. Therefore, we cannot determine who did more jumping jacks initially before the timer started based on the given information.
c. A proportional relationship exists when the ratio of jumping jacks to time remains constant. Looking at the data, we can determine that Kimberly's jumping jacks per second remain constant at 4 jumping jacks per second regardless of the time intervals. Therefore, Kimberly shows a proportional relationship between the number of jumping jacks and time. On the other hand, Katrina's jumping jacks per second vary depending on the time intervals, so she does not show a proportional relationship.
a. To determine which person is doing more jumping jacks per second, we need to calculate the rate of jumping jacks per second for each person.
For Kimberly:
- In 3 seconds, she completed 17 jumping jacks.
- In 8 seconds, she completed 37 jumping jacks.
- In 12 seconds, she completed 53 jumping jacks.
- In 16 seconds, she completed 69 jumping jacks.
To find the rate of jumping jacks per second, we divide the number of jumping jacks by the time in seconds:
- For the first interval: 17/3 ≈ 5.67 jumping jacks per second.
- For the second interval: 37/8 ≈ 4.63 jumping jacks per second.
- For the third interval: 53/12 ≈ 4.42 jumping jacks per second.
- For the fourth interval: 69/16 ≈ 4.31 jumping jacks per second.
For Katrina:
- In 2 seconds, she completed 10 jumping jacks.
- In 5 seconds, she completed 25 jumping jacks.
- In 12 seconds, she completed 60 jumping jacks.
- In 20 seconds, she completed 100 jumping jacks.
To find the rate of jumping jacks per second, we divide the number of jumping jacks by the time in seconds:
- For the first interval: 10/2 = 5 jumping jacks per second.
- For the second interval: 25/5 = 5 jumping jacks per second.
- For the third interval: 60/12 = 5 jumping jacks per second.
- For the fourth interval: 100/20 = 5 jumping jacks per second.
Based on the calculations, both Kimberly and Katrina are doing the same number of jumping jacks per second (5 jumping jacks per second).
b. To determine who has done more jumping jacks initially before the timer started, we need to compare the first data points for each person.
For Kimberly, she completed 17 jumping jacks in 3 seconds.
For Katrina, she completed 10 jumping jacks in 2 seconds.
Therefore, Kimberly has done more jumping jacks initially before the timer started.
c. To determine which person shows a proportional relationship, we need to check if the ratio of jumping jacks to time remains constant.
For Kimberly:
- Between the first and second interval: 17/3 ≈ 5.67 jumping jacks per second
- Between the second and third interval: 37/8 ≈ 4.63 jumping jacks per second
- Between the third and fourth interval: 53/12 ≈ 4.42 jumping jacks per second
For Katrina:
- Between the first and second interval: 10/2 = 5 jumping jacks per second
- Between the second and third interval: 25/5 = 5 jumping jacks per second
- Between the third and fourth interval: 60/12 = 5 jumping jacks per second
Both Kimberly and Katrina show a proportional relationship, as the ratio of jumping jacks to time remains constant for each person.
To answer these questions, you need to compare the number of jumping jacks completed and the time taken by each person. Let's analyze each question step by step:
a. To determine which person is doing more jumping jacks per second, you need to calculate the jumping jacks per second for each person. The formula to find jumping jacks per second is:
Jumping jacks per second = Number of jumping jacks / Time taken (in seconds)
For Kimberly, the jumping jacks per second are as follows:
• At 3 seconds: 17 / 3 ≈ 5.67 jumping jacks per second
• At 8 seconds: 37 / 8 ≈ 4.63 jumping jacks per second
• At 12 seconds: 53 / 12 ≈ 4.42 jumping jacks per second
• At 16 seconds: 69 / 16 ≈ 4.31 jumping jacks per second
For Katrina, the jumping jacks per second are as follows:
• At 2 seconds: 10 / 2 = 5 jumping jacks per second
• At 5 seconds: 25 / 5 = 5 jumping jacks per second
• At 12 seconds: 60 / 12 = 5 jumping jacks per second
• At 20 seconds: 100 / 20 = 5 jumping jacks per second
Comparing the values, both Kimberly and Katrina are doing the same number of jumping jacks per second, which is 5.
b. To determine which person had done more jumping jacks initially before the timer started, you need to look at the number of jumping jacks when the time is zero. In other words, you need to find the value of jumping jacks at the starting time of 0 seconds.
For Kimberly, the jumping jacks at 0 seconds are not provided, so it cannot be determined.
For Katrina, the jumping jacks at 0 seconds are not provided either.
As the data for the initial jumping jacks is missing for both Kimberly and Katrina, we cannot determine who had done more jumping jacks initially.
c. To determine which person shows a proportional relationship, you need to check if the ratio of jumping jacks to time is constant for each person. In other words, if the jumping jacks per second remain the same throughout for a person, then their relationship is proportional.
For Kimberly, the jumping jacks per second are not constant as they decrease gradually.
For Katrina, the jumping jacks per second are constant at 5 jumping jacks per second.
Hence, only Katrina shows a proportional relationship between the number of jumping jacks and the time taken.
In summary:
a. Both Kimberly and Katrina are doing the same number of jumping jacks per second, which is 5.
b. It cannot be determined who had done more jumping jacks initially before the timer started due to missing data.
c. Only Katrina shows a proportional relationship between the number of jumping jacks and the time taken.