Seth is celebrating his sixtieth birthday by vacationing in Europe. He decides to travel around by train, so he can relax and enjoy the scenic views. The cost of a train ride includes a flat fee plus an additional charge per kilometer of the trip.

Question

This table shows the relationship between the distance Seth travels (in kilometers), x, and the cost of the trip (in euro), y.
x (kilometers) y (euro)
40 32
60 41
80 50
100 59
According to the values in the table, do x and y have a proportional relationship?
yes
no
What is the constant of proportionality? Write your answer as a whole number or decimal.
euro per kilometer

Answer 1

To determine if \(x\) (distance in kilometers) and \(y\) (cost in euro) have a proportional relationship, we need to see if the ratio of \(y\) to \(x\) is constant for all entries in the table provided.

We can calculate the cost per kilometer for each entry:

1. For \(x = 40\) km, \(y = 32\) euro: \[ \text{Cost per km} = \frac{32}{40} = 0.8 \text{ euro/km} \]

2. For \(x = 60\) km, \(y = 41\) euro: \[ \text{Cost per km} = \frac{41}{60} \approx 0.6833 \text{ euro/km} \]

3. For \(x = 80\) km, \(y = 50\) euro: \[ \text{Cost per km} = \frac{50}{80} = 0.625 \text{ euro/km} \]

4. For \(x = 100\) km, \(y = 59\) euro: \[ \text{Cost per km} = \frac{59}{100} = 0.59 \text{ euro/km} \]

Since the cost per kilometer is not constant (it varies for each case), \(x\) and \(y\) do not have a proportional relationship.

Therefore, the answer is no, \(x\) and \(y\) do not have a proportional relationship.

As for the constant of proportionality, since there is no constant ratio for all entries, it cannot be defined in this case. Therefore, we can conclude that:

There is no constant of proportionality.