Question
A taxi service charges $3.00 just to get in the cab and $1.00 for each mile traveled.
Could the relationship between the distance traveled and the total cost be a proportional relationship?
Number of miles (m) Cost in dollars (c) Cost per mile (
m
c
)
5 $8.00 1.6
8 $11.00 1.375
20 $23.00 1.15
The relationship could be proportional.
The relationship could not be proportional.
Could the relationship between the distance traveled and the total cost be a proportional relationship?
Number of miles (m) Cost in dollars (c) Cost per mile (
m
c
)
5 $8.00 1.6
8 $11.00 1.375
20 $23.00 1.15
The relationship could be proportional.
The relationship could not be proportional.
Answers
GPT-4o mini
To determine if the relationship between the distance traveled and the total cost is a proportional relationship, we need to consider the definitions of a proportional relationship. A relationship is proportional if it can be expressed as \( y = kx \), where \( k \) is a constant. In such a case, the ratio \( \frac{y}{x} \) (total cost to distance traveled) must be constant for all values of \( x \).
In the case of the taxi service you provided:
1. The total cost (c) is made up of a base fare of $3.00 plus $1.00 per mile traveled (m). The formula for total cost can be expressed as:
\[
c = 3 + 1m
\]
or
\[
c = m + 3
\]
2. The key point here is that there is a fixed starting cost of $3.00. A truly proportional relationship would not have a base charge; it would pass through the origin (0,0) with no initial cost.
Now let’s analyze the numbers you provided:
- For **5 miles**:
\[
c = 3 + 1(5) = 3 + 5 = 8 \, \text{(correct)}
\]
- For **8 miles**:
\[
c = 3 + 1(8) = 3 + 8 = 11 \, \text{(correct)}
\]
- For **20 miles**:
\[
c = 3 + 1(20) = 3 + 20 = 23 \, \text{(correct)}
\]
Now, let’s calculate the cost per mile for the given scenarios:
- For 5 miles:
\[
\frac{8}{5} = 1.6
\]
- For 8 miles:
\[
\frac{11}{8} = 1.375
\]
- For 20 miles:
\[
\frac{23}{20} = 1.15
\]
From these calculations, we can see that the cost per mile decreases as the distance increases, indicating that the total cost does not maintain a consistent ratio with distance.
Thus, the relationship between the distance traveled and total cost **could not be a proportional relationship** because of the initial charge of $3.00 and the changing cost per mile.
### Conclusion:
**The relationship could not be proportional.**
In the case of the taxi service you provided:
1. The total cost (c) is made up of a base fare of $3.00 plus $1.00 per mile traveled (m). The formula for total cost can be expressed as:
\[
c = 3 + 1m
\]
or
\[
c = m + 3
\]
2. The key point here is that there is a fixed starting cost of $3.00. A truly proportional relationship would not have a base charge; it would pass through the origin (0,0) with no initial cost.
Now let’s analyze the numbers you provided:
- For **5 miles**:
\[
c = 3 + 1(5) = 3 + 5 = 8 \, \text{(correct)}
\]
- For **8 miles**:
\[
c = 3 + 1(8) = 3 + 8 = 11 \, \text{(correct)}
\]
- For **20 miles**:
\[
c = 3 + 1(20) = 3 + 20 = 23 \, \text{(correct)}
\]
Now, let’s calculate the cost per mile for the given scenarios:
- For 5 miles:
\[
\frac{8}{5} = 1.6
\]
- For 8 miles:
\[
\frac{11}{8} = 1.375
\]
- For 20 miles:
\[
\frac{23}{20} = 1.15
\]
From these calculations, we can see that the cost per mile decreases as the distance increases, indicating that the total cost does not maintain a consistent ratio with distance.
Thus, the relationship between the distance traveled and total cost **could not be a proportional relationship** because of the initial charge of $3.00 and the changing cost per mile.
### Conclusion:
**The relationship could not be proportional.**