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.

Answers

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.

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