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
A. 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, you can do the following:
1. **Identify the points:** From the graph for Sunnyville Farm and from the table for Willowvale Farm, identify the points that represent the number of apples sold and their corresponding total costs.
2. **Check for proportionality:** A proportional relationship can be identified if the ratio of the total cost to the number of apples sold is constant for every point. This means that if you take any pair of values (number of apples, total cost), the ratio \(\frac{\text{Total Cost}}{\text{Number of Apples}}\) should yield the same value (constant of proportionality) for all pairs.
3. **Calculate the constant of proportionality:** To calculate the constant of proportionality \(k\), use the formula:
\[
k = \frac{\text{Total Cost}}{\text{Number of Apples}}
\]
For example, if Sunnyville Farm sells 10 apples for $20 and Willowvale Farm sells 5 apples for $10, you would calculate:
- Sunnyville: \(k = \frac{20}{10} = 2\)
- Willowvale: \(k = \frac{10}{5} = 2\)
In this scenario, the constant of proportionality \(k\) represents the cost per apple. If \(k = 2\), it means each apple costs $2.
B. To find the equation(s) that can be used to describe the proportional relationship(s):
1. **Identify the linear relationship:** A proportional relationship can be described using the standard form of a straight line equation \(y = kx\), where \(y\) is the total cost, \(x\) is the number of apples, and \(k\) is the constant of proportionality.
2. **Formulate the equation:** Based on the constant of proportionality \(k\) calculated in part A, you can write the equation for Sunnyville Farm as:
\[
\text{Total Cost} = 2 \times (\text{Number of Apples})\]
and for Willowvale Farm as:
\[
\text{Total Cost} = 2 \times (\text{Number of Apples})
\]
3. **General form:** Therefore, if both farms have the same constant of proportionality, the equations for both Sunnyville Farm and Willowvale Farm would be identical, showcasing that they charge the same price per apple. If the prices differ, you would adjust the constant \(k\) accordingly for each farm.
In summary:
- A: Check ratios for proportionality for both farms, calculate \(k\).
- B: Formulate the equation \(y = kx\) for each farm, using the constants found.
1. **Identify the points:** From the graph for Sunnyville Farm and from the table for Willowvale Farm, identify the points that represent the number of apples sold and their corresponding total costs.
2. **Check for proportionality:** A proportional relationship can be identified if the ratio of the total cost to the number of apples sold is constant for every point. This means that if you take any pair of values (number of apples, total cost), the ratio \(\frac{\text{Total Cost}}{\text{Number of Apples}}\) should yield the same value (constant of proportionality) for all pairs.
3. **Calculate the constant of proportionality:** To calculate the constant of proportionality \(k\), use the formula:
\[
k = \frac{\text{Total Cost}}{\text{Number of Apples}}
\]
For example, if Sunnyville Farm sells 10 apples for $20 and Willowvale Farm sells 5 apples for $10, you would calculate:
- Sunnyville: \(k = \frac{20}{10} = 2\)
- Willowvale: \(k = \frac{10}{5} = 2\)
In this scenario, the constant of proportionality \(k\) represents the cost per apple. If \(k = 2\), it means each apple costs $2.
B. To find the equation(s) that can be used to describe the proportional relationship(s):
1. **Identify the linear relationship:** A proportional relationship can be described using the standard form of a straight line equation \(y = kx\), where \(y\) is the total cost, \(x\) is the number of apples, and \(k\) is the constant of proportionality.
2. **Formulate the equation:** Based on the constant of proportionality \(k\) calculated in part A, you can write the equation for Sunnyville Farm as:
\[
\text{Total Cost} = 2 \times (\text{Number of Apples})\]
and for Willowvale Farm as:
\[
\text{Total Cost} = 2 \times (\text{Number of Apples})
\]
3. **General form:** Therefore, if both farms have the same constant of proportionality, the equations for both Sunnyville Farm and Willowvale Farm would be identical, showcasing that they charge the same price per apple. If the prices differ, you would adjust the constant \(k\) accordingly for each farm.
In summary:
- A: Check ratios for proportionality for both farms, calculate \(k\).
- B: Formulate the equation \(y = kx\) for each farm, using the constants found.