Question

1. entrance to a state park costs $6 per vehicle, plus $2 per person. the table shows the entry cost for several recent groups of visitors. is the relationship between the number of people and the total entrance cost a proportional relationship? circle one. yes no explain your thinking.

Explanation:

Step1: Recall the definition of proportional relationship

A proportional relationship has the form \(y = kx\), where \(k\) is the constant of proportionality and \(y\) is zero when \(x = 0\). In the context of cost, the total cost \(C\) should be \(C=kx\) (no additional fixed cost). Here, the cost formula is \(C = 2x+6\) (where \(x\) is the number of people).

Step2: Check the ratio of total cost to number of people

For \(x = 2\), \(\frac{C}{x}=\frac{10}{2}=5\). For \(x = 3\), \(\frac{C}{x}=\frac{12}{3} = 4\). For \(x=4\), \(\frac{C}{x}=\frac{14}{4}=3.5\). For \(x = 10\), \(\frac{C}{x}=\frac{26}{10}=2.6\). The ratios \(\frac{C}{x}\) are not constant. Also, when \(x = 0\) (no people in the vehicle), the cost \(C=6\) (non - zero), which violates the \(y = kx\) (where \(y = 0\) when \(x = 0\)) property of a proportional relationship.

Answer:

No. The relationship is not proportional because the ratio of total cost to number of people is not constant and there is a fixed cost (\( \$6\) per vehicle) which means the cost formula is \(C=2x + 6\) (not of the form \(C=kx\)).