Question

4-51 the choir is planning a trip to the water park. the cost to use a school bus is $350 no matter how many students ride the bus. complete the table at right then graph your result. is this a proportional relationship?
number of students on the trip | bus cost per student ($)
10 |
15 |
20 |
35 |

Explanation:

Step1: Recall Proportional Relationship

A proportional relationship has the form \( y = kx \), where \( k \) is constant, and the graph passes through the origin \((0,0)\). The cost per student here is \( \text{Cost per student} = \frac{\text{Total Bus Cost}}{\text{Number of Students}}=\frac{350}{n} \), where \( n \) is the number of students.

Step2: Analyze the Cost Function

The total bus cost is a fixed \( \$350 \), so the cost per student is \( \frac{350}{n} \). For a proportional relationship, when \( n = 0 \), the cost should be \( 0 \), but here, if \( n = 0 \), the cost is still \( \$350 \) (since the bus cost is fixed). Also, the cost per student changes as \( n \) changes (e.g., for \( n = 10 \), it's \( 35 \); for \( n = 15 \), it's \( \frac{350}{15}\approx23.33 \); for \( n = 20 \), it's \( 17.5 \); for \( n = 35 \), it's \( 10 \)), so the ratio \( \frac{\text{Cost}}{\text{Number of Students}} \) is not constant.

Step3: Check Graph Requirement

A proportional relationship's graph is a straight line through the origin. Here, the cost function is \( C=\frac{350}{n} \) (or \( n=\frac{350}{C} \)), which is a hyperbola, not a straight line through the origin.

Answer:

This is not a proportional relationship because the cost per student (the ratio of total cost to number of students) is not constant, and the relationship does not pass through the origin \((0,0)\) (when there are 0 students, the bus still costs \$350, not \$0).