Let's break down the problem step by step.
Laura is renting a movie with the following cost structure:
- A flat fee of \(\$2.00\)
- An additional fee of \(\$0.50\) for each night she keeps the movie
We need to find the cost function \(c(x)\) that represents the total cost Laura will pay, where \(x\) is the number of nights she keeps the movie.
### Step 1: Understand the Components
1. Fixed fee: Laura always pays \(\$2.00\), regardless of how many nights she rents the movie.
2. Variable fee: Laura pays \(\$0.50\) per night, which will depend on the number of nights \(x\).
### Step 2: Construct the Cost Function
To construct the cost function, we combine both the fixed fee and the variable fee.
#### Fixed Fee Component
The cost incurred from the fixed fee is \(\$2.00\).
#### Variable Fee Component
The cost incurred from the additional fee per night is \( \$0.50 \) multiplied by the number of nights \( x \), giving us \( 0.50x \).
### Step 3: Combine the Two Components
The total cost \( c(x) \), as a function of the number of nights \( x \), is the sum of the fixed fee and the variable fee:
[tex]\[ c(x) = 2.00 + 0.50x \][/tex]
### Step 4: Compare with the Given Choices
Let's compare the constructed cost function with the provided choices:
- Choice 1: \( c(x) = 200x + 50 \)
- This does not match our cost structure.
- Choice 2: \( c(x) = 2.00 + 0.50x \)
- This correctly matches our constructed cost function.
- Choice 3: \( c(x) = 2.50x \)
- This does not correctly separate the fixed fee from the variable fee.
- Choice 4: \( c(x) = (2.00 + 0.50) x \)
- This incorrectly combines the fixed fee and the per-night fee inappropriately.
### Conclusion
The correct cost function that accurately represents the given scenario is:
[tex]\[ c(x) = 2.00 + 0.50x \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{c(x) = 2.00 + 0.50x} \][/tex]
