Question
A teacher driving to a conference in a school van records the number of gallons of gas used and the number of miles traveled.
Complete the table to show the case where there is a proportional relationship between the number of gallons of gas used by the school van and the number of miles traveled.
Gallons of GasSunnyville Farm sells apples according to the graph shown below, and Willowvale Farm sells apples according to the table shown below.

A. Clearly explain how to determine whether one or both farms sell apples following a proportional relationship between the total cost of the apples and the number of apples sold. Also, calculate the constant of proportionality for the proportional relationship(s) and describe its meaning in the context of the scenario.
B. Explain how to find the equation(s) that can be used to describe the proportional relationship(s) in part A.
Type your answer in the box below. Make sure you label your answers for each part.
275
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330
Complete the table to show the case where there is a proportional relationship between the number of gallons of gas used by the school van and the number of miles traveled.
Gallons of GasSunnyville Farm sells apples according to the graph shown below, and Willowvale Farm sells apples according to the table shown below.

A. Clearly explain how to determine whether one or both farms sell apples following a proportional relationship between the total cost of the apples and the number of apples sold. Also, calculate the constant of proportionality for the proportional relationship(s) and describe its meaning in the context of the scenario.
B. Explain how to find the equation(s) that can be used to describe the proportional relationship(s) in part A.
Type your answer in the box below. Make sure you label your answers for each part.
275
ñ
ñ
330
Answers
GPT-4o mini
To analyze the proportional relationship between gas consumption and miles traveled, as well as the cost of apples sold by Sunnyville Farm and Willowvale Farm, follow the steps below:
### Part A:
**Explanation of Proportional Relationships**
To determine if one or both farms sell apples according to a proportional relationship, we look for a constant ratio between the total cost and the number of apples sold. A proportional relationship exists if:
1. For any two points (x, y) where x is the number of apples and y is the total cost, the ratio \( \frac{y}{x} \) is constant.
2. The graph passes through the origin (0, 0), which indicates that if no apples are purchased, the total cost is zero.
**Calculating the Constant of Proportionality**
1. **Sunnyville Farm:**
- If the graph indicates that the cost for 1 apple is \( c \), we would find the constant of proportionality \( k \) as:
\[
k = \frac{\text{Total Cost}}{\text{Number of Apples}}
\]
2. **Willowvale Farm:**
- If the table shows specific pairs of values \( (x, y) \) for number of apples and corresponding costs, calculate \( k \) using the given data in a similar manner.
### Constant of Proportionality in Context:
This constant \( k \) represents the price-per-apple. If \( k = c \), it means that for every apple, the cost increases by \( c \). This indicates how much you will pay for each additional apple, providing a clear understanding of pricing for the customers.
### Part B:
**Finding Equations for the Proportional Relationships**
To write the equations for the proportional relationships:
1. **General Equation:**
\[
y = kx
\]
where \( y \) is the total cost, \( x \) is the number of apples, and \( k \) is the constant of proportionality.
2. **Specific Equations for Each Farm:**
- **Sunnyville Farm:** If \( k \) from the graph is determined to be, say $2.00 per apple, the equation becomes:
\[
y = 2x
\]
- **Willowvale Farm:** If \( k \) is found from the table to be $1.50 per apple, the equation would be:
\[
y = 1.5x
\]
### Conclusion:
The analysis of these apples' selling conditions demonstrates how constant ratios reflect pricing strategies in both farms. Each farm has its own constant of proportionality which allows for easy comparisons and ensures transparency in transactions.
(Note: Specific values and relationships need to be tailored from actual data provided in graphs and tables for complete accuracy; they are illustrative here.)
### Part A:
**Explanation of Proportional Relationships**
To determine if one or both farms sell apples according to a proportional relationship, we look for a constant ratio between the total cost and the number of apples sold. A proportional relationship exists if:
1. For any two points (x, y) where x is the number of apples and y is the total cost, the ratio \( \frac{y}{x} \) is constant.
2. The graph passes through the origin (0, 0), which indicates that if no apples are purchased, the total cost is zero.
**Calculating the Constant of Proportionality**
1. **Sunnyville Farm:**
- If the graph indicates that the cost for 1 apple is \( c \), we would find the constant of proportionality \( k \) as:
\[
k = \frac{\text{Total Cost}}{\text{Number of Apples}}
\]
2. **Willowvale Farm:**
- If the table shows specific pairs of values \( (x, y) \) for number of apples and corresponding costs, calculate \( k \) using the given data in a similar manner.
### Constant of Proportionality in Context:
This constant \( k \) represents the price-per-apple. If \( k = c \), it means that for every apple, the cost increases by \( c \). This indicates how much you will pay for each additional apple, providing a clear understanding of pricing for the customers.
### Part B:
**Finding Equations for the Proportional Relationships**
To write the equations for the proportional relationships:
1. **General Equation:**
\[
y = kx
\]
where \( y \) is the total cost, \( x \) is the number of apples, and \( k \) is the constant of proportionality.
2. **Specific Equations for Each Farm:**
- **Sunnyville Farm:** If \( k \) from the graph is determined to be, say $2.00 per apple, the equation becomes:
\[
y = 2x
\]
- **Willowvale Farm:** If \( k \) is found from the table to be $1.50 per apple, the equation would be:
\[
y = 1.5x
\]
### Conclusion:
The analysis of these apples' selling conditions demonstrates how constant ratios reflect pricing strategies in both farms. Each farm has its own constant of proportionality which allows for easy comparisons and ensures transparency in transactions.
(Note: Specific values and relationships need to be tailored from actual data provided in graphs and tables for complete accuracy; they are illustrative here.)