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Tracy has a cell phone plan that provides 250 free minutes each month for a flat rate of [tex]$\$[/tex]29[tex]$. For any minutes over 250, Tracy is charged $[/tex]\[tex]$0.35$[/tex] per minute.

Which of the following piecewise functions represents charges based on Tracy's cell phone plan?

A. [tex]f(x)=\left\{\begin{array}{c}29, \ x \leq 250 \\ 0.35x, \ x \ \textgreater \ 250\end{array}\right\}[/tex]

B. [tex]f(x)=\left\{\begin{array}{c}29, \ x \leq 250 \\ 29 + 0.35x, \ x \ \textgreater \ 250\end{array}\right\}[/tex]

C. [tex]f(x)=\left\{\begin{array}{c}29, \ x \leq 250 \\ 29 + 0.35(x - 250), \ x \ \textgreater \ 250\end{array}\right\}[/tex]

D. [tex]f(x)=\left\{\begin{array}{c}29, \ x \ \textgreater \ 250 \\ 29 + 35x, \ x \leq 250\end{array}\right\}[/tex]



Answer :

GinnyAnswer
Let's analyze Tracy's cell phone plan step by step to determine the correct piecewise function that represents her monthly charges.

Given:
1. A flat rate of [tex]$29 for up to 250 free minutes. 2. Any additional minutes over the 250 free minutes are charged at $[/tex]0.35 per minute.

We'll break this down into cases based on the number of minutes [tex]\(x\)[/tex] Tracy uses:

Case 1: [tex]\( x \leq 250 \)[/tex]:

If Tracy uses 250 minutes or fewer in a month, no extra charges are incurred. Therefore, the total cost is simply the flat rate:
[tex]\[ f(x) = 29 \][/tex]
for [tex]\(x \leq 250\)[/tex].

Case 2: [tex]\( x > 250 \)[/tex]:

If Tracy uses more than 250 minutes in a month, she will incur additional charges for the extra minutes beyond the 250 free minutes. Specifically:
- The cost for the first 250 minutes is still the flat rate of [tex]$29. - For the additional minutes beyond 250, the cost is $[/tex]0.35 per minute.

Let's denote the additional minutes beyond 250 as [tex]\(x - 250\)[/tex]. The additional cost for these minutes would be:
[tex]\[ 0.35 \times (x - 250) \][/tex]

Therefore, the total cost in this case becomes:
[tex]\[ f(x) = 29 + 0.35 \times (x - 250) \][/tex]
for [tex]\( x > 250 \)[/tex].

Putting both cases together, the piecewise function representing Tracy's monthly charges is:
[tex]\[ f(x) = \begin{cases} 29 & \text{if } x \leq 250 \\ 29 + 0.35 \times (x - 250) & \text{if } x > 250 \end{cases} \][/tex]

Comparing this with the given options, the correct piecewise function is:

C. [tex]\( f(x) = \left\{\begin{array}{c} 29, x \leq 250 \\ 29 + 0.35(x - 250), x > 250 \end{array}\right\} \)[/tex]

So the answer is C.

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