Question
The cost to rent and view a movie online is $2 a month, plus $2 a movie.
y = 2x + 2
To rent a movie and receive it in the mail is $6 a month, plus $1 a movie.
y = x + 6
The costs of the two movie rentals are shown in the graph below.
y = 2x + 2
To rent a movie and receive it in the mail is $6 a month, plus $1 a movie.
y = x + 6
The costs of the two movie rentals are shown in the graph below.
Answers
GPT-4o mini
To analyze the cost of the two movie rental services and understand their graphs, let’s first clarify the equations given:
1. **Online rental service:**
\[
y = 2x + 2
\]
- Here, \(y\) is the total cost, and \(x\) is the number of movies rented.
- The fixed monthly cost is $2, and the variable cost is $2 per movie.
2. **Mail rental service:**
\[
y = x + 6
\]
- Similarly, \(y\) represents the total cost, and \(x\) indicates the number of movies rented.
- The fixed monthly cost is $6, with a variable cost of $1 per movie.
### Comparing the Costs
To understand and compare these two services, we can identify the point at which their costs are equal.
Set the equations equal to each other:
\[
2x + 2 = x + 6
\]
Now, solve for \(x\):
1. Subtract \(x\) from both sides:
\[
2x - x + 2 = 6
\]
\[
x + 2 = 6
\]
2. Subtract 2 from both sides:
\[
x = 4
\]
Now substitute \(x = 4\) into either equation to find \(y\):
Using the online rental service equation:
\[
y = 2(4) + 2 = 8 + 2 = 10
\]
So, when you rent 4 movies, the cost for both services will be $10.
### Summary:
1. Online rental service: Cost is calculated as \(y = 2x + 2\).
2. Mail rental service: Cost is calculated as \(y = x + 6\).
3. Both rental options cost the same ($10) when renting 4 movies.
4. For fewer than 4 movies, the online service is cheaper, while for more than 4 movies, the mail service becomes cheaper.
If you visualize this on a graph:
- The line for the online service would start at (0, 2) with a slope of 2.
- The line for the mail service would start at (0, 6) with a slope of 1.
- The point of intersection at (4, 10) indicates where both services cost the same.
Let me know if you need any further analysis or specific information!
1. **Online rental service:**
\[
y = 2x + 2
\]
- Here, \(y\) is the total cost, and \(x\) is the number of movies rented.
- The fixed monthly cost is $2, and the variable cost is $2 per movie.
2. **Mail rental service:**
\[
y = x + 6
\]
- Similarly, \(y\) represents the total cost, and \(x\) indicates the number of movies rented.
- The fixed monthly cost is $6, with a variable cost of $1 per movie.
### Comparing the Costs
To understand and compare these two services, we can identify the point at which their costs are equal.
Set the equations equal to each other:
\[
2x + 2 = x + 6
\]
Now, solve for \(x\):
1. Subtract \(x\) from both sides:
\[
2x - x + 2 = 6
\]
\[
x + 2 = 6
\]
2. Subtract 2 from both sides:
\[
x = 4
\]
Now substitute \(x = 4\) into either equation to find \(y\):
Using the online rental service equation:
\[
y = 2(4) + 2 = 8 + 2 = 10
\]
So, when you rent 4 movies, the cost for both services will be $10.
### Summary:
1. Online rental service: Cost is calculated as \(y = 2x + 2\).
2. Mail rental service: Cost is calculated as \(y = x + 6\).
3. Both rental options cost the same ($10) when renting 4 movies.
4. For fewer than 4 movies, the online service is cheaper, while for more than 4 movies, the mail service becomes cheaper.
If you visualize this on a graph:
- The line for the online service would start at (0, 2) with a slope of 2.
- The line for the mail service would start at (0, 6) with a slope of 1.
- The point of intersection at (4, 10) indicates where both services cost the same.
Let me know if you need any further analysis or specific information!