A cell phone plan charges 46.75 per month plus  14.06 in taxes, plus 0.30 per minute for calls beyond the 500-minute monthly limit. Write a piecewise-defined function to model the monthly cost c(x) (in $) as a function of the number of minutes used for the month.

To model the monthly cost $ c(x) $ as a function of the number of minutes $ x $ used in a month, we will use a piecewise-defined function. Here's the breakdown:

1. **If the number of minutes used $ x $ is less than or equal to the 500-minute limit**, the cost is just the base monthly charge plus taxes.
2. **If the number of minutes used $ x $ exceeds the 500-minute limit**, there will be an additional charge of $ 0.30 $ per minute for the minutes beyond 500.

Let $ c(x) $ represent the monthly cost.

\[
c(x) = 
\begin{cases} 
46.75 + 14.06 & \text{if } x \leq 500, \\
46.75 + 14.06 + 0.30(x - 500) & \text{if } x > 500.
\end{cases}
\]

### Simplifying:
1. When $ x \leq 500 $, the total cost is $ 46.75 + 14.06 = 60.81 $.
2. When $ x > 500 $, the total cost is $ 60.81 + 0.30(x - 500) $.

Thus, the piecewise function is:

\[
c(x) = 
\begin{cases} 
60.81 & \text{if } x \leq 500, \\
60.81 + 0.30(x - 500) & \text{if } x > 500.
\end{cases}
\]

This function now models the total monthly cost based on the number of minutes $ x $ used.

Would you like further clarification or a more detailed explanation?

### Related Questions:
1. What would be the monthly cost if 450 minutes are used?
2. How much would you pay for using 600 minutes in a month?
3. What is the total additional cost for using 700 minutes?
4. How does the cost function change if the tax amount increases?
5. What happens to the function if the rate for extra minutes changes to $0.40?

### Tip:
When constructing piecewise functions, always consider the behavior of the function at the boundary points, ensuring continuity where necessary.

A cell phone plan charges 46.75 per month plus 14.06 in taxes, plus 0.30 per minute for calls beyond the 500-minute monthly limit. Write a piecewise-defined function to model the monthly cost c(x) (in $) as a function of the number of minutes used for the month.

Learn how to model a cell phone plan with a base charge, taxes, and extra minute charges using a piecewise-defined function. This problem involves algebraic concepts and functions to calculate the total monthly cost for any number of minutes, suitable for students in Grades 9-12.

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