Question
The equation below represents the number of questions Jamie finished in 20 minutes. Let x represent the number of minutes and y represent the number of questions Jamie finishes.
Ashley makes an average of 21 hot dogs at a movie concession stand during her 6-hour shift. Let x represent the number of hours and y represent the number of hot dogs Ashley makes.
Determine which proportional relationship has the greater rate of change of x to y. Then, plot the ordered pairs that represent the rate of change on the provided coordinate plane. Plot only ordered pairs with whole number coordinates.
Ashley makes an average of 21 hot dogs at a movie concession stand during her 6-hour shift. Let x represent the number of hours and y represent the number of hot dogs Ashley makes.
Determine which proportional relationship has the greater rate of change of x to y. Then, plot the ordered pairs that represent the rate of change on the provided coordinate plane. Plot only ordered pairs with whole number coordinates.
Answers
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Answered by
GPT-4o mini
To determine which proportional relationship has the greater rate of change, we need to establish both the rate of change for Jamie and for Ashley's scenarios.
1. **Jamie's Rate of Change:**
- Jamie finished a certain number of questions in 20 minutes. We need to interpret this as a ratio.
- Let’s say Jamie finishes *Q* questions in 20 minutes:
\[
\text{Rate of change (R)} = \frac{\text{Change in y (questions)}}{\text{Change in x (minutes)}} = \frac{Q}{20}
\]
- Without a specific number of questions, we can't determine an exact rate; however, we will denote it as \( R_J = \frac{Q}{20} \).
2. **Ashley’s Rate of Change:**
- Ashley makes an average of 21 hot dogs during her 6-hour shift:
\[
\text{Rate of change (R)} = \frac{\text{Change in y (hot dogs)}}{\text{Change in x (hours)}} = \frac{21}{6} = 3.5
\]
- Thus, Ashley's rate of change is \( R_A = 3.5 \) hot dogs per hour.
**Comparing Rates of Change:**
To compare the rates, we need a value for \( Q \) in Jamie's case. If, for example, Jamie finished 40 questions in 20 minutes, then:
\[
R_J = \frac{40}{20} = 2 \text{ questions per minute} = 120 \text{ questions per hour}
\]
If \( Q = 40 \) questions:
- Jamie's rate of change = 120 questions/hour, which is greater than Ashley's 3.5 hot dogs/hour.
3. **Plotting Ordered Pairs:**
We can create ordered pairs from both scenarios:
**For Jamie** (if she finishes 40 questions in 20 minutes):
- If she finishes 120 questions in 60 minutes, we would consider:
- (0, 0), (20, 40), (40, 80), (60, 120)
**For Ashley**:
- For her average of 21 hot dogs in 6 hours:
- (0, 0), (1, 3.5), (2, 7), (3, 10.5), (4, 14), (5, 17.5), (6, 21)
Since we are plotting only whole-number coordinates, we'll take whole numbers for Jamie:
- Jamie: (0,0), (20,40) (we can scale to whole numbers for hour values if you'd like).
Final coordinates for plotting:
- Jamie: (0, 0), (20, 40)
- Ashley: (0, 0), (6, 21) (but the hour mark may change based on the accuracy expected).
**Overall**:
Assuming Jamie finishes 40 questions as an example, she has a greater rate of change than Ashley. You can graph these ordered pairs on a coordinate plane where the x-axis represents time in minutes/hours and the y-axis represents the number of questions or hot dogs, and you'll observe their respective trends.
1. **Jamie's Rate of Change:**
- Jamie finished a certain number of questions in 20 minutes. We need to interpret this as a ratio.
- Let’s say Jamie finishes *Q* questions in 20 minutes:
\[
\text{Rate of change (R)} = \frac{\text{Change in y (questions)}}{\text{Change in x (minutes)}} = \frac{Q}{20}
\]
- Without a specific number of questions, we can't determine an exact rate; however, we will denote it as \( R_J = \frac{Q}{20} \).
2. **Ashley’s Rate of Change:**
- Ashley makes an average of 21 hot dogs during her 6-hour shift:
\[
\text{Rate of change (R)} = \frac{\text{Change in y (hot dogs)}}{\text{Change in x (hours)}} = \frac{21}{6} = 3.5
\]
- Thus, Ashley's rate of change is \( R_A = 3.5 \) hot dogs per hour.
**Comparing Rates of Change:**
To compare the rates, we need a value for \( Q \) in Jamie's case. If, for example, Jamie finished 40 questions in 20 minutes, then:
\[
R_J = \frac{40}{20} = 2 \text{ questions per minute} = 120 \text{ questions per hour}
\]
If \( Q = 40 \) questions:
- Jamie's rate of change = 120 questions/hour, which is greater than Ashley's 3.5 hot dogs/hour.
3. **Plotting Ordered Pairs:**
We can create ordered pairs from both scenarios:
**For Jamie** (if she finishes 40 questions in 20 minutes):
- If she finishes 120 questions in 60 minutes, we would consider:
- (0, 0), (20, 40), (40, 80), (60, 120)
**For Ashley**:
- For her average of 21 hot dogs in 6 hours:
- (0, 0), (1, 3.5), (2, 7), (3, 10.5), (4, 14), (5, 17.5), (6, 21)
Since we are plotting only whole-number coordinates, we'll take whole numbers for Jamie:
- Jamie: (0,0), (20,40) (we can scale to whole numbers for hour values if you'd like).
Final coordinates for plotting:
- Jamie: (0, 0), (20, 40)
- Ashley: (0, 0), (6, 21) (but the hour mark may change based on the accuracy expected).
**Overall**:
Assuming Jamie finishes 40 questions as an example, she has a greater rate of change than Ashley. You can graph these ordered pairs on a coordinate plane where the x-axis represents time in minutes/hours and the y-axis represents the number of questions or hot dogs, and you'll observe their respective trends.
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