To determine which piecewise function correctly represents the charges based on Tracy's cell phone plan, we need to carefully analyze each option given the details provided:
1. Understand the plan:
- Tracy pays a flat rate of \[tex]$29 for the first 250 minutes.
- For any additional minutes beyond 250, Tracy is charged \$[/tex]0.35 per minute.
2. Analyze each option:
Option A:
[tex]\[
f(x) = \begin{cases}
29, & x > 250 \\
29 + 0.35x, & x \leq 250
\end{cases}
\][/tex]
- For [tex]\(x > 250\)[/tex], it states the charge remains \[tex]$29, which is incorrect because there should be an additional charge for the extra minutes.
- For \(x \leq 250\), it states the charge is \(29 + 0.35x\), which means it incorrectly adds \$[/tex]0.35 per minute for all minutes, even the first 250, which are free.
Option B:
[tex]\[
f(x) = \begin{cases}
29, & x \leq 250 \\
29 + 0.35(x-250), & x > 250
\end{cases}
\][/tex]
- For [tex]\(x \leq 250\)[/tex], it correctly states the charge is a flat \[tex]$29.
- For \(x > 250\), it correctly computes the charge as the flat \$[/tex]29 plus \[tex]$0.35 for each additional minute over 250.
Option C:
\[
f(x) = \begin{cases}
29, & x \leq 250 \\
35x, & x > 250
\end{cases}
\]
- For \(x \leq 250\), it correctly states the charge is \$[/tex]29.
- For [tex]\(x > 250\)[/tex], it incorrectly states the charge is \[tex]$35 per minute for all minutes, which doesn't make sense according to the plan.
Option D:
\[
f(x) = \begin{cases}
29, & x \leq 250 \\
29 + 35x, & x > 250
\end{cases}
\]
- For \(x \leq 250\), it correctly states the charge is \$[/tex]29.
- For [tex]\(x > 250\)[/tex], it incorrectly states the charge is \[tex]$29 plus \$[/tex]35 per minute for all minutes, which is an erroneous interpretation of the plan.
After evaluating each option:
- Option A is incorrect because it miscalculates the charges for both [tex]\(x \leq 250\)[/tex] and [tex]\(x > 250\)[/tex].
- Option C is incorrect because it applies an erroneous \[tex]$35 charge for all minutes when \(x > 250\).
- Option D is incorrect because it adds an excessive \$[/tex]35 per minute charge on top of the flat \$29.
The correct representation is found in Option B, which accurately models the charges:
[tex]\[
f(x) = \begin{cases}
29, & x \leq 250 \\
29 + 0.35(x-250), & x > 250
\end{cases}
\][/tex]
Thus, Option B is the correct piecewise function representing Tracy's cell phone plan charges.
