17. The tables below show the number of jumping jacks completed after a given period of time in seconds.

Question

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?

Answer 1

To answer the questions, we will first calculate the jumping jacks completed per second for each person and analyze the initial values before the timer started.

a. Which person is doing more jumping jacks per second?

Kimberly's Jumping Jacks Rate: We will calculate the rate of jumping jacks per second over each time interval.

1. From 3 seconds to 8 seconds:
   Jumping Jacks = 37 - 17 = 20
   Time = 8 - 3 = 5 seconds
   Rate = 20 Jacks / 5 seconds = 4 Jacks/second

2. From 8 seconds to 12 seconds:
   Jumping Jacks = 53 - 37 = 16
   Time = 12 - 8 = 4 seconds
   Rate = 16 Jacks / 4 seconds = 4 Jacks/second

3. From 12 seconds to 16 seconds:
   Jumping Jacks = 69 - 53 = 16
   Time = 16 - 12 = 4 seconds
   Rate = 16 Jacks / 4 seconds = 4 Jacks/second

Average Rate for Kimberly: 4 Jacks/second

Katrina's Jumping Jacks Rate: Now we will calculate Katrina's rate similarly.

1. From 2 seconds to 5 seconds:
   Jumping Jacks = 25 - 10 = 15
   Time = 5 - 2 = 3 seconds
   Rate = 15 Jacks / 3 seconds = 5 Jacks/second

2. From 5 seconds to 12 seconds:
   Jumping Jacks = 60 - 25 = 35
   Time = 12 - 5 = 7 seconds
   Rate = 35 Jacks / 7 seconds = 5 Jacks/second

3. From 12 seconds to 20 seconds:
   Jumping Jacks = 100 - 60 = 40
   Time = 20 - 12 = 8 seconds
   Rate = 40 Jacks / 8 seconds = 5 Jacks/second

Average Rate for Katrina: 5 Jacks/second

Comparison:
Katrina is doing more jumping jacks per second (5 Jacks/second) compared to Kimberly (4 Jacks/second).

b. Which person had done more jumping jacks initially before the timer started?

Checking the number of jumping jacks completed at the beginning (at 0 seconds):

For Kimberly, there is no provided initial value, so we assume it to be 0 since all values start from 3 seconds.

For Katrina, the initial value is given as 10 jumping jacks at 2 seconds.

Comparison:
Katrina had done more jumping jacks initially (10) compared to Kimberly (0).

c. Which person shows a proportional relationship?

A proportional relationship exists if the ratios of jumping jacks to time remain constant.

Kimberly's data:

• At 3 seconds: $\frac{17}{3}$
• At 8 seconds: $\frac{37}{8}$
• At 12 seconds: $\frac{53}{12}$
• At 16 seconds: $\frac{69}{16}$

Calculating the ratios:

• $\frac{17}{3}\approx 5.67$
• $\frac{37}{8}=4.625$
• $\frac{53}{12}\approx 4.42$
• $\frac{69}{16}=4.3125$

(These ratios are not consistent)

Katrina's data:

• At 2 seconds: $\frac{10}{2}=5$
• At 5 seconds: $\frac{25}{5}=5$
• At 12 seconds: $\frac{60}{12}=5$
• At 20 seconds: $\frac{100}{20}=5$

(Since all ratios equal to 5, this shows a consistent ratio)

Conclusion:
Katrina shows a proportional relationship, while Kimberly does not.

Summary of Answers:

a. Katrina is doing more jumping jacks per second.
b. Katrina had done more jumping jacks initially before the timer started.
c. Katrina shows a proportional relationship.