Answer :
To solve this problem, let's carefully break down the cell phone plan's billing structure and then construct the corresponding graph. Here's the step-by-step walkthrough:
### Step 1: Understand the Billing Structure
1. Base Cost:
- Arionna's plan costs [tex]$\$[/tex] 20[tex]$ per month for up to 300 minutes of calls. 2. Overage Charges: - If Arionna uses more than 300 minutes, she incurs an additional $[/tex]\[tex]$ 5$[/tex] overage fee.
- Beyond the 300 minutes, there is also a charge of [tex]$\$[/tex] 0.25$ per minute.
### Step 2: Construct the Cost Function
- Let [tex]\( x \)[/tex] be the number of minutes Arionna uses in a month.
- The cost [tex]\( y \)[/tex] in dollars will have two cases based on whether [tex]\( x \)[/tex] is within the included minutes or exceeds it.
#### Case 1: [tex]\( x \leq 300 \)[/tex]
For [tex]\( x \)[/tex] up to and including 300 minutes, Arionna only pays the base cost:
[tex]\[ y = 20 \][/tex]
#### Case 2: [tex]\( x > 300 \)[/tex]
For [tex]\( x \)[/tex] exceeding 300 minutes, Arionna pays the base cost plus the overage fee and the additional per-minute charge for each minute beyond 300:
[tex]\[ y = 20 + 5 + 0.25(x - 300) \][/tex]
[tex]\[ y = 25 + 0.25(x - 300) \][/tex]
Simplifying further:
[tex]\[ y = 25 + 0.25x - 75 \][/tex]
[tex]\[ y = 0.25x - 50 \][/tex]
### Step 3: Piece Together the Function
Combining both cases, the function for the monthly cost [tex]\( y \)[/tex] based on [tex]\( x \)[/tex] minutes is:
[tex]\[ y = \begin{cases} 20 & \text{if } 0 \leq x \leq 300 \\ 0.25x - 50 & \text{if } x > 300 \end{cases} \][/tex]
### Step 4: Graph the Function
- For [tex]\( x \)[/tex] from [tex]\( 0 \)[/tex] to [tex]\( 300 \)[/tex]: The graph is a horizontal line at [tex]\( y = 20 \)[/tex].
- For [tex]\( x > 300 \)[/tex]: The graph is a line with a slope of [tex]\( 0.25 \)[/tex] starting from the point [tex]\( (300, 20) \)[/tex].
We start at [tex]\( (300, 20) \)[/tex]:
- At [tex]\( x = 300 \)[/tex], [tex]\( y = 20 \)[/tex].
- Then as [tex]\( x \)[/tex] increases by 1 minute, [tex]\( y \)[/tex] increases by [tex]\( 0.25 \)[/tex].
### Visual Representation
Below is a simulated range and their calculated costs:
- [tex]\( x \)[/tex] ranges from [tex]\( 0 \)[/tex] to [tex]\( 600 \)[/tex] minutes with [tex]\( 1000 \)[/tex] points in-between.
- The graph [tex]\( y \)[/tex] values are segmented where all minutes up to [tex]\( 300 \)[/tex] have the cost [tex]\( y = 20 \)[/tex] and beyond that, the cost starts increasing linearly.
For illustration purposes, here are the selected values across the range:
[tex]\[ x = [0, 0.6006006, 1.2012012, ..., 599.3993994, 600] \][/tex]
[tex]\[ y = [20, 20, 20, ..., 99.54954955, 100] \][/tex]
- Notice for [tex]\( x \leq 300 \)[/tex]: [tex]\( y = 20 \)[/tex].
- For [tex]\( x > 300 \)[/tex], cost increases like [tex]\( y = 25 + 0.25(x - 300) \)[/tex].
### Graphical Representation:
When plotting the function:
1. From [tex]\( x = 0 \)[/tex] to [tex]\( x = 300 \)[/tex], the line will be straight and horizontal at [tex]\( y = 20 \)[/tex].
2. From [tex]\( x = 300 \)[/tex] onwards, the line will slope upwards starting linearly from [tex]\( y = 20 \)[/tex].
This visual represents how the monthly cost changes with the number of minutes used, adhering to Arionna's plan structure.
### Step 1: Understand the Billing Structure
1. Base Cost:
- Arionna's plan costs [tex]$\$[/tex] 20[tex]$ per month for up to 300 minutes of calls. 2. Overage Charges: - If Arionna uses more than 300 minutes, she incurs an additional $[/tex]\[tex]$ 5$[/tex] overage fee.
- Beyond the 300 minutes, there is also a charge of [tex]$\$[/tex] 0.25$ per minute.
### Step 2: Construct the Cost Function
- Let [tex]\( x \)[/tex] be the number of minutes Arionna uses in a month.
- The cost [tex]\( y \)[/tex] in dollars will have two cases based on whether [tex]\( x \)[/tex] is within the included minutes or exceeds it.
#### Case 1: [tex]\( x \leq 300 \)[/tex]
For [tex]\( x \)[/tex] up to and including 300 minutes, Arionna only pays the base cost:
[tex]\[ y = 20 \][/tex]
#### Case 2: [tex]\( x > 300 \)[/tex]
For [tex]\( x \)[/tex] exceeding 300 minutes, Arionna pays the base cost plus the overage fee and the additional per-minute charge for each minute beyond 300:
[tex]\[ y = 20 + 5 + 0.25(x - 300) \][/tex]
[tex]\[ y = 25 + 0.25(x - 300) \][/tex]
Simplifying further:
[tex]\[ y = 25 + 0.25x - 75 \][/tex]
[tex]\[ y = 0.25x - 50 \][/tex]
### Step 3: Piece Together the Function
Combining both cases, the function for the monthly cost [tex]\( y \)[/tex] based on [tex]\( x \)[/tex] minutes is:
[tex]\[ y = \begin{cases} 20 & \text{if } 0 \leq x \leq 300 \\ 0.25x - 50 & \text{if } x > 300 \end{cases} \][/tex]
### Step 4: Graph the Function
- For [tex]\( x \)[/tex] from [tex]\( 0 \)[/tex] to [tex]\( 300 \)[/tex]: The graph is a horizontal line at [tex]\( y = 20 \)[/tex].
- For [tex]\( x > 300 \)[/tex]: The graph is a line with a slope of [tex]\( 0.25 \)[/tex] starting from the point [tex]\( (300, 20) \)[/tex].
We start at [tex]\( (300, 20) \)[/tex]:
- At [tex]\( x = 300 \)[/tex], [tex]\( y = 20 \)[/tex].
- Then as [tex]\( x \)[/tex] increases by 1 minute, [tex]\( y \)[/tex] increases by [tex]\( 0.25 \)[/tex].
### Visual Representation
Below is a simulated range and their calculated costs:
- [tex]\( x \)[/tex] ranges from [tex]\( 0 \)[/tex] to [tex]\( 600 \)[/tex] minutes with [tex]\( 1000 \)[/tex] points in-between.
- The graph [tex]\( y \)[/tex] values are segmented where all minutes up to [tex]\( 300 \)[/tex] have the cost [tex]\( y = 20 \)[/tex] and beyond that, the cost starts increasing linearly.
For illustration purposes, here are the selected values across the range:
[tex]\[ x = [0, 0.6006006, 1.2012012, ..., 599.3993994, 600] \][/tex]
[tex]\[ y = [20, 20, 20, ..., 99.54954955, 100] \][/tex]
- Notice for [tex]\( x \leq 300 \)[/tex]: [tex]\( y = 20 \)[/tex].
- For [tex]\( x > 300 \)[/tex], cost increases like [tex]\( y = 25 + 0.25(x - 300) \)[/tex].
### Graphical Representation:
When plotting the function:
1. From [tex]\( x = 0 \)[/tex] to [tex]\( x = 300 \)[/tex], the line will be straight and horizontal at [tex]\( y = 20 \)[/tex].
2. From [tex]\( x = 300 \)[/tex] onwards, the line will slope upwards starting linearly from [tex]\( y = 20 \)[/tex].
This visual represents how the monthly cost changes with the number of minutes used, adhering to Arionna's plan structure.
