- Identify the linear cost model: \(c(x) = px + s\), where \(p\) is the price per ticket and \(s\) is the service fee.
- Use the given data \((s = 5.50\) and \(c(5) = 108.00)\) to set up the equation \(108.00 = 5p + 5.50\).
- Solve for the price per ticket, finding \(p = 20.50\).
- The resulting cost function is \(c(x) = 5.50 + 20.50x\), which corresponds to the first option.
- Final Answer: \(\boxed{c(x) = 5.50 + 20.50x}\)
Explanation
- Analyzing the Problem Components
The problem asks to find the linear function \(c(x)\) representing the total cost for purchasing \(x\) tickets. We are given that there is a fixed service fee of \(\$5.50\) and a constant price per ticket. The total cost for 5 tickets is \(\$108.00\). The general linear form is \(c(x) = px + s\), where \(p\) is the price per ticket, \(s\) is the service fee, and \(x\) is the number of tickets.
- Solving for the Price per Ticket
From the problem description, we know the service fee is \(s = 5.50\). Substituting this into the general linear equation, we get:
\[c(x) = px + 5.50\]
We are also given the information that for 5 tickets, the total cost is \(\$108.00\), which can be written as:
\[c(5) = 108.00\]
By substituting \(x = 5\) into our equation, we obtain:
\[108.00 = p(5) + 5.50\]
Now, we solve for \(p\):
\[108.00 - 5.50 = 5p\]
\[102.50 = 5p\]
\[p = \frac{102.50}{5} = 20.50\]
The price per ticket is \(\$20.50\).
- Determining the Correct Function
With the price per ticket \(p = 20.50\) and the service fee \(s = 5.50\), the linear function for the total cost is:
\[c(x) = 20.50x + 5.50\]
Comparing this result with the given options, we find it matches the first choice: \(c(x) = 5.50 + 20.50x\).
Examples
Linear cost models are used daily in various services. For example, if you rent a car for a daily rate plus a one-time insurance fee, your total cost can be modeled as \(C(d) = rd + f\), where \(d\) is the number of days, \(r\) is the daily rate, and \(f\) is the fixed insurance fee. Similarly, a plumber might charge a flat call-out fee plus an hourly rate. Understanding how to construct these functions allows you to calculate total expenses for any service based on usage or quantity.