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The problem in the image involves Parisa's cell phone plan and has two parts:

**Given:**
- The monthly fee for Parisa's plan is $95, which includes unlimited free text messages, phone calls, and **5 GB** of data.
- Each additional gigabyte of data (or part of a gigabyte) costs **$8**.

### Part a: Find the cost of a month in which Parisa used **6.3 GB** of data.

To calculate this:
1. Parisa’s plan includes **5 GB** of data, so the additional data used is:
   \[
   6.3 \, \text{GB} - 5 \, \text{GB} = 1.3 \, \text{GB}
   \]
   
2. Since each additional gigabyte (or part of a gigabyte) costs **$8**, Parisa will pay for **2 full gigabytes** (because any part of a gigabyte is rounded up to the next whole GB).

Thus, the extra cost is:
\[
2 \times 8 = 16 \, \text{dollars}
\]

3. The total cost for the month will be the base plan fee plus the extra cost:
\[
95 \, \text{dollars} + 16 \, \text{dollars} = 111 \, \text{dollars}
\]

### Part b: Express the monthly charge $ C(x) $ as a piecewise function, where $ x $ represents the number of gigabytes used.

The monthly charge function $ C(x) $ can be written as:

\[
C(x) = 
\begin{cases} 
95 & \text{if } 0 \leq x \leq 5 \\
95 + 8 \cdot \lceil x - 5 \rceil & \text{if } x > 5 
\end{cases}
\]

Where $ \lceil x - 5 \rceil $ is the ceiling function, which rounds up the excess data beyond 5 GB to the next whole number of gigabytes.

---

Would you like further explanation or details?

Here are 5 related questions to expand on this:
1. How would the cost change if Parisa used **7.8 GB** of data instead?
2. What would the total cost be if the plan only included **4 GB** of data instead of **5 GB**?
3. How would the function $ C(x) $ change if the additional data cost was reduced to **$5** per gigabyte?
4. Could you calculate the total cost for someone who used **10.2 GB** of data in a month?
5. What would happen to the monthly charge if no extra data was used?

**Tip**: When dealing with piecewise functions, always pay attention to boundaries, and remember to correctly handle conditions like rounding up with functions like the ceiling function.

Parisa's cell phone plan has a monthly fee of $95, which includes unlimited free text messages and phone calls and 5GB of data. Each extra gigabyte of data, or part thereof, costs $8.
a. Find the cost of a month in which Parisa used 6.3GB of data.
b. If x represents the number of gigabytes used, express the monthly charge C(x) of this plan as a piecewise function.

Learn how to calculate the cost of a cell phone plan with a base fee of $95 and extra charges for data usage. Explore how to use piecewise functions and ceiling functions to find the monthly charge when using more than the included 5GB of data. Suitable for high school students studying algebra and functions.

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