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?
a. To determine which person is doing more jumping jacks per second, we need to calculate the jacks per second for each person. The jacks per second is found by dividing the number of jumping jacks by the time in seconds.
For Kimberly:
3 seconds - 17 jumping jacks, so 17 jacks / 3 sec = 5.67 jacks/sec
8 seconds - 37 jumping jacks, so 37 jacks / 8 sec = 4.63 jacks/sec
12 seconds - 53 jumping jacks, so 53 jacks / 12 sec = 4.42 jacks/sec
16 seconds - 69 jumping jacks, so 69 jacks / 16 sec = 4.31 jacks/sec
For Katrina:
2 seconds - 10 jumping jacks, so 10 jacks / 2 sec = 5 jacks/sec
5 seconds - 25 jumping jacks, so 25 jacks / 5 sec = 5 jacks/sec
12 seconds - 60 jumping jacks, so 60 jacks / 12 sec = 5 jacks/sec
20 seconds - 100 jumping jacks, so 100 jacks / 20 sec = 5 jacks/sec
From the calculations, we can see that both Kimberly and Katrina are doing the same number of jumping jacks per second, which is 5 jacks/sec.
b. To determine which person had done more jumping jacks initially before the timer started, we need to look at the first data point for each person.
For Kimberly, the first data point is at 3 seconds, and she completed 17 jumping jacks.
For Katrina, the first data point is at 2 seconds, and she completed 10 jumping jacks.
Therefore, Kimberly 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 need to compare the ratios of jumping jacks to time for each person. If the ratios are constant, then there is a proportional relationship.
For Kimberly:
17 jacks / 3 sec = 5.67 jacks/sec
37 jacks / 8 sec = 4.63 jacks/sec
53 jacks / 12 sec = 4.42 jacks/sec
69 jacks / 16 sec = 4.31 jacks/sec
The ratios for Kimberly are not constant, indicating that there is no proportional relationship.
For Katrina:
10 jacks / 2 sec = 5 jacks/sec
25 jacks / 5 sec = 5 jacks/sec
60 jacks / 12 sec = 5 jacks/sec
100 jacks / 20 sec = 5 jacks/sec
The ratios for Katrina are constant, indicating that there is a proportional relationship between time and jumping jacks for Katrina.
a. To determine which person is doing more jumping jacks per second, we need to calculate their respective rates. The rate is the number of jumping jacks divided by the time in seconds.
For Kimberly:
- Rate = (Number of jumping jacks for a given time) / (Time in seconds)
For time = 3 seconds, Kimberly completed 17 jumping jacks. So her rate for 3 seconds is:
- Kimberly's rate for 3 seconds = 17/3 = 5.67 jumping jacks per second (rounded to two decimal places)
Similarly, we can calculate Kimberly's rates for 8, 12, and 16 seconds:
- Kimberly's rate for 8 seconds = 37/8 = 4.63 jumping jacks per second (rounded to two decimal places)
- Kimberly's rate for 12 seconds = 53/12 = 4.42 jumping jacks per second (rounded to two decimal places)
- Kimberly's rate for 16 seconds = 69/16 = 4.31 jumping jacks per second (rounded to two decimal places)
For Katrina:
- Katrina's rate for 2 seconds = 10/2 = 5 jumping jacks per second
- Katrina's rate for 5 seconds = 25/5 = 5 jumping jacks per second
- Katrina's rate for 12 seconds = 60/12 = 5 jumping jacks per second
- Katrina's rate for 20 seconds = 100/20 = 5 jumping jacks per second
Comparing the rates, we can see that both Kimberly and Katrina are doing the same number of jumping jacks per second. Therefore, both persons are doing an equal number of jumping jacks per second.
b. To determine which person had done more jumping jacks initially before the timer started, we need to look at their values when time is zero. In this case, for both Kimberly and Katrina, there is no given value for zero seconds. Therefore, we can't determine who had done more jumping jacks initially before the timer was started.
c. To identify a proportional relationship, we need to examine whether the rates for each person are constant and if they have a constant ratio. In this case, both Kimberly and Katrina have rates that are not constant. As the time changes, their rates change as well. Therefore, there is no proportional relationship between the number of jumping jacks and the time for either Kimberly or Katrina.
To compare the number of jumping jacks per second, we need to calculate the rate of jumping jacks for each person by dividing the number of jumping jacks by the time in seconds.
a. Kimberly's jumping jack rate:
For Kimberly, we divide the number of jumping jacks by the time in seconds:
- 17 jumping jacks / 3 seconds = 5.67 jumping jacks per second
- 37 jumping jacks / 8 seconds = 4.63 jumping jacks per second
- 53 jumping jacks / 12 seconds = 4.42 jumping jacks per second
- 69 jumping jacks / 16 seconds = 4.31 jumping jacks per second
b. Katrina's jumping jack rate:
Similarly, for Katrina, we divide the number of jumping jacks by the time in seconds:
- 10 jumping jacks / 2 seconds = 5 jumping jacks per second
- 25 jumping jacks / 5 seconds = 5 jumping jacks per second
- 60 jumping jacks / 12 seconds = 5 jumping jacks per second
- 100 jumping jacks / 20 seconds = 5 jumping jacks per second
By comparing the rates, we can see that both Kimberly and Katrina complete the same number of jumping jacks per second, which is 5 jumping jacks.
b. To determine who has completed more jumping jacks initially before the timer started, we need to look at the data when the timer is at its lowest value.
For Kimberly:
- At 3 seconds, Kimberly completed 17 jumping jacks.
For Katrina:
- At 2 seconds, Katrina completed 10 jumping jacks.
By comparing the initial jumping jacks completed before the timer started, we can see that Kimberly completed more jumping jacks initially with 17 jumping jacks.
c. A proportional relationship exists when the rate of change between the two quantities is constant. To determine if there is a proportional relationship between time and the number of jumping jacks for both Kimberly and Katrina, we need to compare the rates for both individuals.
For Kimberly, there is a slight decrease in the rate of jumping jacks per second as time increases. This means that the relationship between time and jumping jacks is not exactly proportional.
For Katrina, the rate of jumping jacks per second remains constant at 5 jumping jacks for all time intervals. This indicates a proportional relationship between time and jumping jacks for Katrina.
In conclusion:
a. Both Kimberly and Katrina are doing the same number of jumping jacks per second, which is 5 jumping jacks.
b. Kimberly had done more jumping jacks initially before the timer started with 17 jumping jacks.
c. Katrina shows a proportional relationship between time and the number of jumping jacks.