A cell phone plan charges $20 per month for unlimited call and text messages, however each gigabyte (GB) of data is charged at a rate of $8 per GB. Assume that fractional values of a GB are prorated according to this rate. If a user uses more than 10 GB of data during their monthly billing cycle, then they are not charged more than the cost for the first 10 GB.

Write a piecewise-defined function that describes the amount a cell phone user will pay, C , as a function of the number of GB of data they use, d , over the course of a month.

Do not include the dollar sign in any of your answers.

To write the piecewise-defined function for the amount a cell phone user will pay, $ C $, as a function of the number of GB of data they use, $ d $, over the course of a month, we need to consider the given conditions:

1. The base cost is $20 per month for unlimited calls and text messages.
2. Each gigabyte (GB) of data is charged at $8 per GB.
3. If a user uses more than 10 GB, the cost is capped at the price for 10 GB of data.

Let's define the function step-by-step:

### 1. When the data usage is 10 GB or less ($ d \leq 10 $):
- The total cost $ C $ consists of the base cost ($20) plus $8 for each GB used.
- Therefore, for $ d \leq 10 $, the cost function can be written as:
  \[
  C(d) = 20 + 8d
  \]

### 2. When the data usage is more than 10 GB ($ d > 10 $):
- The cost is capped at the cost for 10 GB of data.
- This cost is calculated as:
  \[
  C(d) = 20 + 8 \times 10 = 20 + 80 = 100
  \]
- Therefore, for $ d > 10 $, the cost function is:
  \[
  C(d) = 100
  \]

### Piecewise-Defined Function

Combining these conditions, the piecewise-defined function for the cost $ C(d) $ is:

\[
C(d) = 
\begin{cases} 
20 + 8d & \text{if } 0 \leq d \leq 10 \\
100 & \text{if } d > 10 
\end{cases}
\]

This function effectively models the user's total cost based on their data usage in GB over a month.

### Would you like to go into more details, or have any questions?

### Related Questions:
1. What would be the cost for 5 GB of data usage?
2. How much would the user pay for 12 GB of data usage?
3. How is the piecewise function helpful in analyzing different scenarios of data usage?
4. How would the cost change if the price per GB increased to $10?
5. How would you graph the piecewise function $ C(d) $ over the interval $ 0 \leq d \leq 20 $?

### Tip:
Piecewise functions are particularly useful for modeling real-life situations where rules or conditions change based on different scenarios.

A cell phone plan charges $20 per month for unlimited call and text messages, however each gigabyte (GB) of data is charged at a rate of $8 per GB. Assume that fractional values of a GB are prorated according to this rate. If a user uses more than 10 GB of data during their monthly billing cycle, then they are not charged more than the cost for the first 10 GB.

Write a piecewise-defined function that describes the amount a cell phone user will pay, C, as a function of the number of GB of data they use, d, over the course of a month.

Learn how to create a piecewise-defined function that models a cell phone plan charging $20 per month for calls and texts, and $8 per GB of data with a 10 GB cap. This problem covers key concepts such as linear equations and piecewise functions, suitable for students in Grades 8-10.

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