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 not be proportional.

The relationship could be proportional.

Answer 1

To determine if the relationship between the distance traveled and the total cost of the taxi service is proportional, we need to define what a proportional relationship is. A relationship is proportional if it can be expressed as \( c = km \), where \( k \) is a constant of proportionality, and the graph of the relationship is a straight line through the origin (0,0).

In this case, the base fare is $3.00 (the cost when no miles are traveled), and the charge per mile is $1.00. The total cost can be expressed as:

\[ c = 3 + 1m \]

This means that for any given amount of miles traveled, the cost does not start from $0, because there is a fixed base fare of $3.00 added to the cost regardless of the miles traveled.

Let's see the calculations:

- For 5 miles:
\[ c = 3 + 1(5) = 8 \]

- For 8 miles:
\[ c = 3 + 1(8) = 11 \]

- For 20 miles:
\[ c = 3 + 1(20) = 23 \]

Now, calculating the cost per mile (which is given in your table):

1. For 5 miles:
\[ \frac{8}{5} = 1.6 \]

2. For 8 miles:
\[ \frac{11}{8} \approx 1.375 \]

3. For 20 miles:
\[ \frac{23}{20} = 1.15 \]

As you can see, the cost per mile is not constant; it varies based on the distance traveled. This indicates that the relationship between the distance traveled and the total cost is not proportional, because a proportional relationship would require that the cost per mile remains constant, which is not the case here.

Therefore, the answer is:

**The relationship could not be proportional.**