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.
number of apples 15 the cost is 3.10
number of apples 20 the cost is 3.80
number of apples 25 the cost of it is 5.20
number of apples 30 the cost of it is 5.20
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. Determining Proportional Relationships and Constant of Proportionality**
To determine whether one or both farms sell apples following a proportional relationship, we need to check if the ratio of total cost to the number of apples sold is constant.
**Sunnyville Farm:**
We do not have a specific graph provided; however, we will assume it is linear based on typical sales data. A linear relationship could indicate a proportional relationship if it passes through the origin (0,0).
**Willowvale Farm:**
Using the provided data:
- For 15 apples: Cost = $3.10 → Ratio = 3.10 / 15 = 0.2067
- For 20 apples: Cost = $3.80 → Ratio = 3.80 / 20 = 0.1900
- For 25 apples: Cost = $5.20 → Ratio = 5.20 / 25 = 0.2080
- For 30 apples: Cost = $5.20 → Ratio = 5.20 / 30 = 0.1733
Now we analyze the ratios:
- The ratio for 15 apples is approximately 0.2067,
- The ratio for 20 apples is 0.1900,
- The ratio for 25 apples is approximately 0.2080,
- The ratio for 30 apples is approximately 0.1733.
Since these ratios change and are not consistent, Willowvale Farm does not follow a proportional relationship.
**Constant of Proportionality:**
If Sunnyville Farm were confirmed to follow a proportional relationship and if we had actual data showing a constant ratio (for example, if we identified that it consistently yielded a fixed ratio like 0.25), that ratio would serve as the constant of proportionality. It signifies the cost per apple when apples are sold proportionally.
**B. Finding the Equation(s) to Describe the Proportional Relationship(s)**
For a proportional relationship, the general equation is of the form:
\[ y = kx \]
where:
- \( y \) is the total cost,
- \( x \) is the number of apples sold,
- \( k \) is the constant of proportionality.
1. **Sunnyville Farm (Hypothetical Proportional Relationship):**
If we find \( k \) (the constant of proportionality) through the graph data (assuming it shows consistent cost-to-number ratios), we could say:
\[ y = kx \]
For example, if we found \( k = 0.25 \), the equation would be:
\[ y = 0.25x \]
This would mean each apple costs $0.25.
2. **Willowvale Farm:**
As there is no proportional relationship, we cannot derive a linear equation. Thus, additional equations or data points showing specific costs for certain quantities must be used to describe their relationships in segments rather than as a simple proportional model.
In conclusion, only Sunnyville could potentially have a proportional relationship, represented by a linear equation based on the determined constant of proportionality \( k \), while Willowvale Farm does not maintain a proportional relationship based on varying ratios.
To determine whether one or both farms sell apples following a proportional relationship, we need to check if the ratio of total cost to the number of apples sold is constant.
**Sunnyville Farm:**
We do not have a specific graph provided; however, we will assume it is linear based on typical sales data. A linear relationship could indicate a proportional relationship if it passes through the origin (0,0).
**Willowvale Farm:**
Using the provided data:
- For 15 apples: Cost = $3.10 → Ratio = 3.10 / 15 = 0.2067
- For 20 apples: Cost = $3.80 → Ratio = 3.80 / 20 = 0.1900
- For 25 apples: Cost = $5.20 → Ratio = 5.20 / 25 = 0.2080
- For 30 apples: Cost = $5.20 → Ratio = 5.20 / 30 = 0.1733
Now we analyze the ratios:
- The ratio for 15 apples is approximately 0.2067,
- The ratio for 20 apples is 0.1900,
- The ratio for 25 apples is approximately 0.2080,
- The ratio for 30 apples is approximately 0.1733.
Since these ratios change and are not consistent, Willowvale Farm does not follow a proportional relationship.
**Constant of Proportionality:**
If Sunnyville Farm were confirmed to follow a proportional relationship and if we had actual data showing a constant ratio (for example, if we identified that it consistently yielded a fixed ratio like 0.25), that ratio would serve as the constant of proportionality. It signifies the cost per apple when apples are sold proportionally.
**B. Finding the Equation(s) to Describe the Proportional Relationship(s)**
For a proportional relationship, the general equation is of the form:
\[ y = kx \]
where:
- \( y \) is the total cost,
- \( x \) is the number of apples sold,
- \( k \) is the constant of proportionality.
1. **Sunnyville Farm (Hypothetical Proportional Relationship):**
If we find \( k \) (the constant of proportionality) through the graph data (assuming it shows consistent cost-to-number ratios), we could say:
\[ y = kx \]
For example, if we found \( k = 0.25 \), the equation would be:
\[ y = 0.25x \]
This would mean each apple costs $0.25.
2. **Willowvale Farm:**
As there is no proportional relationship, we cannot derive a linear equation. Thus, additional equations or data points showing specific costs for certain quantities must be used to describe their relationships in segments rather than as a simple proportional model.
In conclusion, only Sunnyville could potentially have a proportional relationship, represented by a linear equation based on the determined constant of proportionality \( k \), while Willowvale Farm does not maintain a proportional relationship based on varying ratios.