https://file.aiquickdraw.com/mathbot/file/1726534530993ixynb5xb.jpg

Let's break down the problem and work through the parts:

### Given Information:
- **Monthly charge** = $30
- **Free call minutes** = 200 minutes
- **Charge after free minutes** = $0.03 per minute (after 200 minutes)

#### (a) Function notation
We can define the cost function $ C(t) $, where $ t $ is the number of minutes used.

- If $ t \leq 200 $, the customer is within the free minutes, so the cost is just $30.
  
- If $ t > 200 $, the cost will be $30 plus $0.03 for each minute over 200.

Thus, the function can be written as:
\[
C(t) = \begin{cases} 
30 & \text{if } t \leq 200 \\
30 + 0.03(t - 200) & \text{if } t > 200
\end{cases}
\]

#### (b) Cost for 350 minutes
For $ t = 350 $ minutes:

Since 350 minutes is greater than 200 minutes, we use the second part of the function:
\[
C(350) = 30 + 0.03(350 - 200) = 30 + 0.03 \times 150 = 30 + 4.50 = 34.50
\]
The cost for 350 minutes is **$34.50**.

#### (c) Cost for 180 minutes
For $ t = 180 $ minutes:

Since 180 minutes is within the free 200 minutes, we use the first part of the function:
\[
C(180) = 30
\]
The cost for 180 minutes is **$30**.

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Would you like a more detailed explanation of any of these steps?

Here are some related questions you might consider:
1. What would the cost be for using exactly 200 minutes?
2. How does the cost change if you use 250 minutes?
3. If the rate after 200 minutes changed to $0.05 per minute, how would that affect the function?
4. What would the cost function look like if the company offered only 150 free minutes instead of 200?
5. How would you represent the total minutes as a graph, based on the cost function?

**Tip**: When defining piecewise functions, always check which condition (range) applies before calculating the values!

A telephone company charges $30 a month and gives the customer 200 free call minutes. After the 200 min, the company charges $0.03 a minute. 
(a) Write the function using function notation. 
(b) Find the cost for talking 350 min in a month. 
(c) Find the cost for talking 180 min in a month.

Explore how to model a telephone company's monthly charges with a piecewise function. The company offers 200 free minutes and charges $0.03 per minute afterward. Learn how to calculate costs for 350 minutes and 180 minutes, ideal for Grade 9-10 students.

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