17. The tables below show the number of jumping jacks completed after a given period of time in seconds.
Kimberly:
Time (seconds) Jumping Jacks
3 17
8 37
12 53
16 69
Katrina:
Time (seconds) Jumping Jacks
2 10
5 25
12 60
20 100(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?
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a. To determine which person is doing more jumping jacks per second, we can divide the total number of jumping jacks by the total time in seconds for each person.
For Kimberly, the total time is 16 seconds and the total number of jumping jacks is 69.
Jacks per second for Kimberly = 69/16 = 4.3125
For Katrina, the total time is 20 seconds and the total number of jumping jacks is 100.
Jacks per second for Katrina = 100/20 = 5
Therefore, Katrina is doing more jumping jacks per second.
b. To determine which person had done more jumping jacks initially before the timer started, we need to look at the number of jumping jacks at time 0.
For Kimberly, there is no information about the number of jumping jacks at time 0.
For Katrina, there is information about the number of jumping jacks at time 2 seconds, which is 10.
Therefore, Katrina had done more jumping jacks initially before the timer started.
c. To determine which person shows a proportional relationship between time and jumping jacks, we can look at the ratios of jumping jacks to time for each person.
For Kimberly, the ratios are:
17/3 = 5.67
37/8 = 4.625
53/12 = 4.42
69/16 = 4.3125
For Katrina, the ratios are:
10/2 = 5
25/5 = 5
60/12 = 5
100/20 = 5
Both Kimberly and Katrina show a constant ratio of 5, so they both show a proportional relationship between time and jumping jacks.
a. To determine which person is doing more jumping jacks per second, we can calculate the rate of jumping jacks per second for each person.
For Kimberly, we can calculate the jumping jacks per second by dividing the number of jumping jacks by the time in seconds:
3 seconds: 17 jumping jacks / 3 seconds = 5.67 jumping jacks per second
8 seconds: 37 jumping jacks / 8 seconds = 4.62 jumping jacks per second
12 seconds: 53 jumping jacks / 12 seconds = 4.42 jumping jacks per second
16 seconds: 69 jumping jacks / 16 seconds = 4.31 jumping jacks per second
For Katrina, we can calculate the jumping jacks per second in the same way:
2 seconds: 10 jumping jacks / 2 seconds = 5 jumping jacks per second
5 seconds: 25 jumping jacks / 5 seconds = 5 jumping jacks per second
12 seconds: 60 jumping jacks / 12 seconds = 5 jumping jacks per second
20 seconds: 100 jumping jacks / 20 seconds = 5 jumping jacks per second
From the calculations, we can see that both Kimberly and Katrina are doing the same amount of jumping jacks per second, which is 5 jumping jacks per second.
b. To determine which person had done more jumping jacks initially before the timer started, we can look at the table. The table shows that Kimberly did 17 jumping jacks at 3 seconds, while Katrina did 10 jumping jacks at 2 seconds.
Therefore, initially before the timer started, Kimberly had done more jumping jacks.
c. A proportional relationship exists when the ratio of jumping jacks to time is the same for each person.
To determine if there is a proportional relationship, we can compare the ratios of jumping jacks to time for each person:
Kimberly:
3 seconds: 17 jumping jacks / 3 seconds = 5.67 jumping jacks per second
8 seconds: 37 jumping jacks / 8 seconds = 4.62 jumping jacks per second
12 seconds: 53 jumping jacks / 12 seconds = 4.42 jumping jacks per second
16 seconds: 69 jumping jacks / 16 seconds = 4.31 jumping jacks per second
Katrina:
2 seconds: 10 jumping jacks / 2 seconds = 5 jumping jacks per second
5 seconds: 25 jumping jacks / 5 seconds = 5 jumping jacks per second
12 seconds: 60 jumping jacks / 12 seconds = 5 jumping jacks per second
20 seconds: 100 jumping jacks / 20 seconds = 5 jumping jacks per second
From the calculations, we can see that the ratios of jumping jacks to time are the same for each person, which indicates a proportional relationship.
To answer these questions, we need to calculate the rate of jumping jacks per second for each person and compare the initial number of jumping jacks.
a. To find the rate of jumping jacks per second for each person, divide the total number of jumping jacks by the total time taken in seconds. Let's calculate it for each person:
Kimberly:
Rate = Total Jumping Jacks / Total Time
Rate = 69 / 16
Rate = 4.3125 jumping jacks/second
Katrina:
Rate = Total Jumping Jacks / Total Time
Rate = 100 / 20
Rate = 5 jumping jacks/second
Therefore, Katrina is doing more jumping jacks per second compared to Kimberly.
b. To find the initial number of jumping jacks, we need to look at the first entry of each person's data:
Kimberly: 17 Jumping Jacks initially
Katrina: 10 Jumping Jacks initially
Therefore, Kimberly had done more jumping jacks initially before the timer started.
c. To determine if there is a proportional relationship, we need to check if the rate of jumping jacks per second is constant for each person. Let's calculate the rates for different time intervals for both individuals:
Kimberly:
Rate for 3 seconds = 17 / 3 = 5.67 jumping jacks/second
Rate for 8 seconds = 37 / 8 = 4.625 jumping jacks/second
Rate for 12 seconds = 53 / 12 = 4.4167 jumping jacks/second
Rate for 16 seconds = 69 / 16 = 4.3125 jumping jacks/second
Katrina:
Rate for 2 seconds = 10 / 2 = 5 jumping jacks/second
Rate for 5 seconds = 25 / 5 = 5 jumping jacks/second
Rate for 12 seconds = 60 / 12 = 5 jumping jacks/second
Rate for 20 seconds = 100 / 20 = 5 jumping jacks/second
Since the rate of jumping jacks per second remains constant for both persons, they both show a proportional relationship.
Therefore, the answers are:
a. Katrina is doing more jumping jacks per second.
b. Kimberly had done more jumping jacks initially before the timer started.
c. Both persons show a proportional relationship.