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.

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 this question, we will break it down into two parts: Part A and Part B.
### Part A: Determining Proportional Relationships and Constant of Proportionality
To determine whether one or both farms sell apples according to a proportional relationship between the total cost and the number of apples sold, follow these steps:
1. **Examine the Graph and Table**:
- For Sunnyville Farm, you would analyze the points plotted on the graph, looking for a straight line that passes through the origin (0,0). This indicates a proportional relationship.
- For Willowvale Farm, review the values in the table. Calculate the total cost for each quantity of apples sold to see if the ratio of total cost to the number of apples sold remains constant.
2. **Calculate the Constant of Proportionality**:
- If Sunnyville Farm's graph shows a straight line through the origin, calculate the slope (rise over run) between two points on the line to determine the constant of proportionality (k). The formula is:
\[
k = \frac{\text{Total Cost}}{\text{Number of Apples}}
\]
- For Willowvale Farm, calculate the ratio of total cost to the number of apples from the table for multiple entries. If the ratio remains the same for all entries, the value is the constant of proportionality.
**Meaning in Context**:
The constant of proportionality (k) represents the unit price of the apples. For example, if k = 2, this means that each apple costs $2.
### Part B: Finding the Equation of the Proportional Relationship(s)
To find the equation that can be used to describe the proportional relationships identified in part A, use the following approach:
1. **Formulate the Equation**:
- For both farms, the proportional relationship can be expressed with the equation:
\[
y = kx
\]
where:
- **y** is the total cost,
- **k** is the constant of proportionality (unit price),
- **x** is the number of apples.
2. **Determine the Specific Equation(s)**:
- For Sunnyville Farm's graph, substitute the constant of proportionality derived from the slope into the equation.
- For Willowvale Farm, use the constant of proportionality calculated from the table.
**Example**:
If Sunnyville Farm has a constant of proportionality of k = 2, the equation would be:
\[
y = 2x
\]
indicating that the cost increases by $2 for each apple sold.
If Willowvale Farm has the same constant of proportionality, its equation would be:
\[
y = 2x
\]
In summary, both farms can be described by the same equation if they have the same constant of proportionality, indicating both sell apples at the same price per apple.
### Summary:
- **Part A**: Check for a straight line through the origin. Calculate k for both farms. The value of k indicates the cost per apple.
- **Part B**: Use the k value in the equation \( y = kx \) to describe the proportional relationships.
### Part A: Determining Proportional Relationships and Constant of Proportionality
To determine whether one or both farms sell apples according to a proportional relationship between the total cost and the number of apples sold, follow these steps:
1. **Examine the Graph and Table**:
- For Sunnyville Farm, you would analyze the points plotted on the graph, looking for a straight line that passes through the origin (0,0). This indicates a proportional relationship.
- For Willowvale Farm, review the values in the table. Calculate the total cost for each quantity of apples sold to see if the ratio of total cost to the number of apples sold remains constant.
2. **Calculate the Constant of Proportionality**:
- If Sunnyville Farm's graph shows a straight line through the origin, calculate the slope (rise over run) between two points on the line to determine the constant of proportionality (k). The formula is:
\[
k = \frac{\text{Total Cost}}{\text{Number of Apples}}
\]
- For Willowvale Farm, calculate the ratio of total cost to the number of apples from the table for multiple entries. If the ratio remains the same for all entries, the value is the constant of proportionality.
**Meaning in Context**:
The constant of proportionality (k) represents the unit price of the apples. For example, if k = 2, this means that each apple costs $2.
### Part B: Finding the Equation of the Proportional Relationship(s)
To find the equation that can be used to describe the proportional relationships identified in part A, use the following approach:
1. **Formulate the Equation**:
- For both farms, the proportional relationship can be expressed with the equation:
\[
y = kx
\]
where:
- **y** is the total cost,
- **k** is the constant of proportionality (unit price),
- **x** is the number of apples.
2. **Determine the Specific Equation(s)**:
- For Sunnyville Farm's graph, substitute the constant of proportionality derived from the slope into the equation.
- For Willowvale Farm, use the constant of proportionality calculated from the table.
**Example**:
If Sunnyville Farm has a constant of proportionality of k = 2, the equation would be:
\[
y = 2x
\]
indicating that the cost increases by $2 for each apple sold.
If Willowvale Farm has the same constant of proportionality, its equation would be:
\[
y = 2x
\]
In summary, both farms can be described by the same equation if they have the same constant of proportionality, indicating both sell apples at the same price per apple.
### Summary:
- **Part A**: Check for a straight line through the origin. Calculate k for both farms. The value of k indicates the cost per apple.
- **Part B**: Use the k value in the equation \( y = kx \) to describe the proportional relationships.