https://file.aiquickdraw.com/mathbot/file/172740449823860zn2d1q.jpg

Let's break down and address the questions from the image:

**(a) Find a linear function that expresses the monthly bill $ B(m) $ as a function of minutes used $ m $:**
   The linear function given in the question is:

   \[
   B(m) = 0.01m + 5.17
   \]

   This equation means that the monthly bill $ B(m) $ is $5.17 plus $0.01 per minute used.

**(b) What are the independent and dependent variables?**
   - The independent variable is $ m $, which represents the **number of minutes used**.
   - The dependent variable is $ B(m) $, which represents the **monthly bill**.

**(c) What is the implied domain of this linear function for a 31-day month?**
   - The number of minutes used in a 31-day month can range from 0 minutes to the maximum possible minutes in 31 days.
   - Maximum number of minutes in 31 days = $ 31 \times 24 \times 60 = 44,640 $ minutes.
   - Therefore, the domain is $ [0, 44,640] $.

**(d) What is the monthly bill if 1,297 minutes are used for long-distance calls?**
   - Plug $ m = 1,297 $ into the equation $ B(m) = 0.01m + 5.17 $:

   \[
   B(1,297) = 0.01(1,297) + 5.17 = 12.97 + 5.17 = 18.14
   \]

   The monthly bill would be **$18.14**.

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If you'd like to dive deeper into any of these steps, let me know! Here are five questions to consider further:

1. What does the slope (0.01) represent in the linear equation?
2. How would the equation change if the cost per minute was $0.05 instead?
3. What would the monthly bill be if no minutes were used?
4. How would the maximum possible bill change for a month with 30 days?
5. How can we graph this function, and what would the key points look like?

**Tip:** In linear equations, the slope is often a rate of change, showing how the dependent variable changes with respect to the independent variable.

A phone company has a long-distance phone plan that charges a monthly fee of $5.17 plus $0.01 per minute. Answer the following questions: (a) Find a linear function that expresses the monthly bill as a function of minutes used. (b) What are the independent and dependent variables? (c) What is the implied domain of the linear function for a 31-day month? (d) What is the monthly bill if 1,297 minutes are used?

Explore how to find a linear function that calculates the monthly bill for a long-distance phone plan, using the equation B(m) = 0.01m + 5.17. Learn about dependent and independent variables, and calculate the bill for 1,297 minutes of use. Suitable for students in Grades 8-10.

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