Question

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?

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.