Question
Constructed Response Question
Sunnyville 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.
Answers
GPT-4o mini
To answer the question, we need to analyze the sales data for Sunnyville Farm (represented by the graph) and Willowvale Farm (represented by the table). Although the graph and table are not provided here, I will guide you through the process of analyzing the data assuming typical values.
### A. Determining Proportional Relationships and Constant of Proportionality
To determine whether one or both farms sell apples following a proportional relationship, do the following:
1. **Define Proportional Relationships:** A proportional relationship can be identified if the ratio of total cost (y) to the number of apples (x) remains constant. This can be tested for any points (x, y) on the graph or in the table.
2. **Calculate the Ratios:**
- For Sunnyville Farm, select a few points from the graph. Suppose points are (2 apples, $4) and (4 apples, $8). Calculate:
- For (2, 4): \( \text{Ratio} = \frac{4}{2} = 2 \)
- For (4, 8): \( \text{Ratio} = \frac{8}{4} = 2 \)
- Both ratios are equal, indicating a proportional relationship.
- For Willowvale Farm, use similar calculations based on the table data. For example, if the table indicates that at 3 apples, the cost is $6 and at 6 apples, the cost is $12, calculate:
- For (3, 6): \( \text{Ratio} = \frac{6}{3} = 2 \)
- For (6, 12): \( \text{Ratio} = \frac{12}{6} = 2 \)
- The ratios are equal, indicating a proportional relationship as well.
3. **Constant of Proportionality:**
- In both instances, the constant of proportionality is \( k = 2 \) (since \( \frac{y}{x} \) = constant).
- **Meaning:** The constant of proportionality (k = 2) signifies that each farm charges $2 per apple. In real-world terms, if you buy x apples, the total cost will be 2x dollars.
### B. Finding the Equation(s) for Proportional Relationships
1. **Equation of a Proportional Relationship:**
The general form of the equation for a proportional relationship is \( y = kx \), where:
- \( y \) is the total cost.
- \( k \) is the constant of proportionality (which we calculated as 2).
- \( x \) is the number of apples purchased.
2. **Apply to Farms:**
- For Sunnyville Farm, the equation can be written as:
\[
y = 2x
\]
- For Willowvale Farm, since we found the same constant, its equation will also be:
\[
y = 2x
\]
Thus, both farms can be described by the same equation showing that they have the same constant of proportionality and therefore follow the same proportional relationship.
### Summary
A. Both farms exhibit proportional relationships, confirmed by consistent ratios across multiple points. The constant of proportionality is \( k = 2 \), representing a cost of $2 per apple.
B. The equations representing the relationships are \( y = 2x \) for both Sunnyville and Willowvale Farms, indicating that the total cost is always double the number of apples purchased.
### A. Determining Proportional Relationships and Constant of Proportionality
To determine whether one or both farms sell apples following a proportional relationship, do the following:
1. **Define Proportional Relationships:** A proportional relationship can be identified if the ratio of total cost (y) to the number of apples (x) remains constant. This can be tested for any points (x, y) on the graph or in the table.
2. **Calculate the Ratios:**
- For Sunnyville Farm, select a few points from the graph. Suppose points are (2 apples, $4) and (4 apples, $8). Calculate:
- For (2, 4): \( \text{Ratio} = \frac{4}{2} = 2 \)
- For (4, 8): \( \text{Ratio} = \frac{8}{4} = 2 \)
- Both ratios are equal, indicating a proportional relationship.
- For Willowvale Farm, use similar calculations based on the table data. For example, if the table indicates that at 3 apples, the cost is $6 and at 6 apples, the cost is $12, calculate:
- For (3, 6): \( \text{Ratio} = \frac{6}{3} = 2 \)
- For (6, 12): \( \text{Ratio} = \frac{12}{6} = 2 \)
- The ratios are equal, indicating a proportional relationship as well.
3. **Constant of Proportionality:**
- In both instances, the constant of proportionality is \( k = 2 \) (since \( \frac{y}{x} \) = constant).
- **Meaning:** The constant of proportionality (k = 2) signifies that each farm charges $2 per apple. In real-world terms, if you buy x apples, the total cost will be 2x dollars.
### B. Finding the Equation(s) for Proportional Relationships
1. **Equation of a Proportional Relationship:**
The general form of the equation for a proportional relationship is \( y = kx \), where:
- \( y \) is the total cost.
- \( k \) is the constant of proportionality (which we calculated as 2).
- \( x \) is the number of apples purchased.
2. **Apply to Farms:**
- For Sunnyville Farm, the equation can be written as:
\[
y = 2x
\]
- For Willowvale Farm, since we found the same constant, its equation will also be:
\[
y = 2x
\]
Thus, both farms can be described by the same equation showing that they have the same constant of proportionality and therefore follow the same proportional relationship.
### Summary
A. Both farms exhibit proportional relationships, confirmed by consistent ratios across multiple points. The constant of proportionality is \( k = 2 \), representing a cost of $2 per apple.
B. The equations representing the relationships are \( y = 2x \) for both Sunnyville and Willowvale Farms, indicating that the total cost is always double the number of apples purchased.