Question
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?
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?
Answers
GPT-4o mini
To analyze the data provided, let's calculate the rate of jumping jacks per second for Kimberly and Katrina, examine their initial jumping jacks, and determine if there's a proportional relationship.
### a. Jumping jacks per second
**Kimberly:**
1. For the 3 seconds:
\( \frac{17 \, \text{jacks}}{3 \, \text{seconds}} \approx 5.67 \, \text{jacks/second} \)
2. For the 8 seconds:
\( \frac{37 \, \text{jacks}}{8 \, \text{seconds}} = 4.625 \, \text{jacks/second} \)
3. For the 12 seconds:
\( \frac{53 \, \text{jacks}}{12 \, \text{seconds}} \approx 4.42 \, \text{jacks/second} \)
4. For the 16 seconds:
\( \frac{69 \, \text{jacks}}{16 \, \text{seconds}} \approx 4.31 \, \text{jacks/second} \)
**Average jacks per second for Kimberly:**
- Total jacks: 17 + 37 + 53 + 69 = 176
- Total seconds: 3 + 8 + 12 + 16 = 39
- Average: \( \frac{176 \, \text{jacks}}{39 \, \text{seconds}} \approx 4.51 \, \text{jacks/second} \)
**Katrina:**
1. For the 2 seconds:
\( \frac{10 \, \text{jacks}}{2 \, \text{seconds}} = 5 \, \text{jacks/second} \)
2. For the 5 seconds:
\( \frac{25 \, \text{jacks}}{5 \, \text{seconds}} = 5 \, \text{jacks/second} \)
3. For the 12 seconds:
\( \frac{60 \, \text{jacks}}{12 \, \text{seconds}} = 5 \, \text{jacks/second} \)
4. For the 20 seconds:
\( \frac{100 \, \text{jacks}}{20 \, \text{seconds}} = 5 \, \text{jacks/second} \)
**Average jacks per second for Katrina:**
- Total jacks: 10 + 25 + 60 + 100 = 195
- Total seconds: 2 + 5 + 12 + 20 = 39
- Average: \( \frac{195 \, \text{jacks}}{39 \, \text{seconds}} = 5 \, \text{jacks/second} \)
**Conclusion for (a):**
- Kimberly averaged around 4.51 jacks/second.
- Katrina had a steady rate of 5 jacks/second.
- **Katrina is doing more jumping jacks per second.**
### b. Initial jumping jacks
To determine the initial jumping jacks before the timer started, we can look at the data provided:
- For **Kimberly**, all values start from time 3 seconds (17 jumping jacks), hence she started with 0 before she began.
- For **Katrina**, the first recorded value is at 2 seconds with 10 jumping jacks, so she had completed 10 jumping jacks before the timer started.
**Conclusion for (b):**
- **Katrina had done more jumping jacks initially before the timer started.** (10 vs. 0)
### c. Proportional relationship
A proportional relationship means that the ratio of jumping jacks to time remains constant.
- For **Kimberly**, the ratio fluctuates. For example:
- \( 17 / 3 \approx 5.67 \)
- \( 37 / 8 = 4.625 \)
- \( 53 / 12 \approx 4.42 \)
- \( 69 / 16 \approx 4.31 \)
Because the ratios are not constant, this does not represent a proportional relationship.
- For **Katrina**, every time point yields a ratio of 5 \( \text{jacks/second} \):
- \( 10 / 2 = 5 \)
- \( 25 / 5 = 5 \)
- \( 60 / 12 = 5 \)
- \( 100 / 20 = 5 \)
Since Katrina has a constant ratio of jumping jacks to time, **she shows a proportional relationship.**
**Conclusion for (c):**
- **Katrina shows a proportional relationship.**
### a. Jumping jacks per second
**Kimberly:**
1. For the 3 seconds:
\( \frac{17 \, \text{jacks}}{3 \, \text{seconds}} \approx 5.67 \, \text{jacks/second} \)
2. For the 8 seconds:
\( \frac{37 \, \text{jacks}}{8 \, \text{seconds}} = 4.625 \, \text{jacks/second} \)
3. For the 12 seconds:
\( \frac{53 \, \text{jacks}}{12 \, \text{seconds}} \approx 4.42 \, \text{jacks/second} \)
4. For the 16 seconds:
\( \frac{69 \, \text{jacks}}{16 \, \text{seconds}} \approx 4.31 \, \text{jacks/second} \)
**Average jacks per second for Kimberly:**
- Total jacks: 17 + 37 + 53 + 69 = 176
- Total seconds: 3 + 8 + 12 + 16 = 39
- Average: \( \frac{176 \, \text{jacks}}{39 \, \text{seconds}} \approx 4.51 \, \text{jacks/second} \)
**Katrina:**
1. For the 2 seconds:
\( \frac{10 \, \text{jacks}}{2 \, \text{seconds}} = 5 \, \text{jacks/second} \)
2. For the 5 seconds:
\( \frac{25 \, \text{jacks}}{5 \, \text{seconds}} = 5 \, \text{jacks/second} \)
3. For the 12 seconds:
\( \frac{60 \, \text{jacks}}{12 \, \text{seconds}} = 5 \, \text{jacks/second} \)
4. For the 20 seconds:
\( \frac{100 \, \text{jacks}}{20 \, \text{seconds}} = 5 \, \text{jacks/second} \)
**Average jacks per second for Katrina:**
- Total jacks: 10 + 25 + 60 + 100 = 195
- Total seconds: 2 + 5 + 12 + 20 = 39
- Average: \( \frac{195 \, \text{jacks}}{39 \, \text{seconds}} = 5 \, \text{jacks/second} \)
**Conclusion for (a):**
- Kimberly averaged around 4.51 jacks/second.
- Katrina had a steady rate of 5 jacks/second.
- **Katrina is doing more jumping jacks per second.**
### b. Initial jumping jacks
To determine the initial jumping jacks before the timer started, we can look at the data provided:
- For **Kimberly**, all values start from time 3 seconds (17 jumping jacks), hence she started with 0 before she began.
- For **Katrina**, the first recorded value is at 2 seconds with 10 jumping jacks, so she had completed 10 jumping jacks before the timer started.
**Conclusion for (b):**
- **Katrina had done more jumping jacks initially before the timer started.** (10 vs. 0)
### c. Proportional relationship
A proportional relationship means that the ratio of jumping jacks to time remains constant.
- For **Kimberly**, the ratio fluctuates. For example:
- \( 17 / 3 \approx 5.67 \)
- \( 37 / 8 = 4.625 \)
- \( 53 / 12 \approx 4.42 \)
- \( 69 / 16 \approx 4.31 \)
Because the ratios are not constant, this does not represent a proportional relationship.
- For **Katrina**, every time point yields a ratio of 5 \( \text{jacks/second} \):
- \( 10 / 2 = 5 \)
- \( 25 / 5 = 5 \)
- \( 60 / 12 = 5 \)
- \( 100 / 20 = 5 \)
Since Katrina has a constant ratio of jumping jacks to time, **she shows a proportional relationship.**
**Conclusion for (c):**
- **Katrina shows a proportional relationship.**