Question
At a restaurant, hot chocolate can be purchased in two different cup sizes. A 12-ounce cup costs $4.80 and a 16-ounce cup costs $6.40.
Is the linear relationship between cup size and cost also proportional? Why or why not?
Is the linear relationship between cup size and cost also proportional? Why or why not?
Answers
GPT-4o mini
To determine if the relationship between cup size and cost is proportional, we need to examine the relationship between the two variables—cup size (in ounces) and cost (in dollars).
For a relationship to be proportional, it must satisfy two criteria:
1. The ratio between the two variables must be constant.
2. The relationship must pass through the origin (0,0), meaning that a 0-ounce cup would cost $0.
Let's calculate the cost per ounce for each cup size:
1. **12-ounce cup**:
- Cost = $4.80
- Cost per ounce = \( \frac{4.80}{12} = 0.40 \) dollars per ounce.
2. **16-ounce cup**:
- Cost = $6.40
- Cost per ounce = \( \frac{6.40}{16} = 0.40 \) dollars per ounce.
Since both cups have the same cost per ounce (0.40), it suggests that there is a linear relationship with a constant rate of change.
However, we also need to consider whether this relationship passes through the origin. If a cup of hot chocolate is sized at 0 ounces, it should theoretically cost $0, but we have not tested this because there is no information about a 0-ounce cup.
Thus, the relationship is linear because the cost per ounce is constant. However, it is **not proportional** because we do not have an origin point defined (0, 0) in the context of cup sizes and costs. The costs do not account for the possibility of a 0-ounce cup costing $0, as a 0-ounce portion does not exist in practical terms in this context.
In conclusion, while the relationship between cup size and cost is linear, it is **not proportional** due to the absence of a defined point at the origin (0,0).
For a relationship to be proportional, it must satisfy two criteria:
1. The ratio between the two variables must be constant.
2. The relationship must pass through the origin (0,0), meaning that a 0-ounce cup would cost $0.
Let's calculate the cost per ounce for each cup size:
1. **12-ounce cup**:
- Cost = $4.80
- Cost per ounce = \( \frac{4.80}{12} = 0.40 \) dollars per ounce.
2. **16-ounce cup**:
- Cost = $6.40
- Cost per ounce = \( \frac{6.40}{16} = 0.40 \) dollars per ounce.
Since both cups have the same cost per ounce (0.40), it suggests that there is a linear relationship with a constant rate of change.
However, we also need to consider whether this relationship passes through the origin. If a cup of hot chocolate is sized at 0 ounces, it should theoretically cost $0, but we have not tested this because there is no information about a 0-ounce cup.
Thus, the relationship is linear because the cost per ounce is constant. However, it is **not proportional** because we do not have an origin point defined (0, 0) in the context of cup sizes and costs. The costs do not account for the possibility of a 0-ounce cup costing $0, as a 0-ounce portion does not exist in practical terms in this context.
In conclusion, while the relationship between cup size and cost is linear, it is **not proportional** due to the absence of a defined point at the origin (0,0).