Question
A movie streaming service charges a monthly fee of $4.50 and an additional fee of $0.35 for every movie downloaded. Now, the total monthly fee is represented by the linear function f(x) = 0.35x + 4.50, where x is the number of movies downloaded in a month.
Answers
GPT-4o mini
The function \( f(x) = 0.35x + 4.50 \) represents the total monthly fee for using the movie streaming service, where:
- \( f(x) \) is the total monthly fee.
- \( x \) is the number of movies downloaded in a month.
- The term \( 0.35x \) represents the additional cost of downloading \( x \) movies, with each movie costing $0.35.
- The fixed fee of $4.50 is the monthly subscription charge, regardless of how many movies are downloaded.
### Key Points to Note:
1. **Fixed Monthly Fee**: Even if no movies are downloaded, the user must pay $4.50.
2. **Variable Cost**: For every movie downloaded, there is an additional cost of $0.35.
3. **Total Cost Calculation**:
- If a user downloads \( x \) movies, the total cost can be calculated by substituting \( x \) into the function \( f(x) \).
- For example, if a user downloads 10 movies:
\[
f(10) = 0.35(10) + 4.50 = 3.50 + 4.50 = 8.00
\]
Therefore, the total monthly fee would be $8.00.
### Graphing the Function:
- The y-intercept of the graph occurs at \( (0, 4.50) \).
- The slope of the line is \( 0.35 \), indicating that for each additional movie downloaded, the total fee increases by $0.35.
- The graph would be a straight line starting from \( (0, 4.50) \) and increasing.
### Example Usage:
1. **If no movies are downloaded (\( x = 0 \))**:
\[
f(0) = 0.35(0) + 4.50 = 4.50
\]
2. **If 5 movies are downloaded (\( x = 5 \))**:
\[
f(5) = 0.35(5) + 4.50 = 1.75 + 4.50 = 6.25
\]
3. **If 20 movies are downloaded (\( x = 20 \))**:
\[
f(20) = 0.35(20) + 4.50 = 7.00 + 4.50 = 11.50
\]
This linear function allows users to easily estimate their monthly fees based on their downloading habits.
- \( f(x) \) is the total monthly fee.
- \( x \) is the number of movies downloaded in a month.
- The term \( 0.35x \) represents the additional cost of downloading \( x \) movies, with each movie costing $0.35.
- The fixed fee of $4.50 is the monthly subscription charge, regardless of how many movies are downloaded.
### Key Points to Note:
1. **Fixed Monthly Fee**: Even if no movies are downloaded, the user must pay $4.50.
2. **Variable Cost**: For every movie downloaded, there is an additional cost of $0.35.
3. **Total Cost Calculation**:
- If a user downloads \( x \) movies, the total cost can be calculated by substituting \( x \) into the function \( f(x) \).
- For example, if a user downloads 10 movies:
\[
f(10) = 0.35(10) + 4.50 = 3.50 + 4.50 = 8.00
\]
Therefore, the total monthly fee would be $8.00.
### Graphing the Function:
- The y-intercept of the graph occurs at \( (0, 4.50) \).
- The slope of the line is \( 0.35 \), indicating that for each additional movie downloaded, the total fee increases by $0.35.
- The graph would be a straight line starting from \( (0, 4.50) \) and increasing.
### Example Usage:
1. **If no movies are downloaded (\( x = 0 \))**:
\[
f(0) = 0.35(0) + 4.50 = 4.50
\]
2. **If 5 movies are downloaded (\( x = 5 \))**:
\[
f(5) = 0.35(5) + 4.50 = 1.75 + 4.50 = 6.25
\]
3. **If 20 movies are downloaded (\( x = 20 \))**:
\[
f(20) = 0.35(20) + 4.50 = 7.00 + 4.50 = 11.50
\]
This linear function allows users to easily estimate their monthly fees based on their downloading habits.