To determine which piecewise function accurately represents Ben's cell phone plan charges, let's break down the conditions and charges described.
1. Ben pays a flat rate of \$39 for up to 200 minutes each month.
2. For any minutes over 200, he is charged an additional \$0.35 per minute.
We need to construct the piecewise function \( f(x) \) that encompasses this information, where \( x \) represents the total number of minutes used in a month.
Step-by-Step Solution:
- If \( x \leq 200 \):
Ben uses 200 minutes or less, so the cost is simply the flat rate of \$39. Thus:
[tex]\[
f(x) = 39 \quad \text{for } x \leq 200
\][/tex]
- If \( x > 200 \):
Ben uses more than 200 minutes, so in addition to the flat rate of \[tex]$39, he has to pay an extra \$[/tex]0.35 for each minute over 200. For instance, if he uses \( x \) minutes, the number of minutes over 200 is \( x - 200 \). Therefore, the additional charge is:
[tex]\[
\text{Additional charge} = 0.35 \times (x - 200)
\][/tex]
Hence, the total cost in this case is:
[tex]\[
f(x) = 39 + 0.35 \times (x - 200) \quad \text{for } x > 200
\][/tex]
Putting this together, the piecewise function should be written as:
[tex]\[
f(x) = \begin{cases}
39, & \text{if } x \leq 200 \\
39 + 0.35(x - 200), & \text{if } x > 200
\end{cases}
\][/tex]
Comparing this with the given options:
- Option A matches this piecewise function exactly:
[tex]\[
f(x) = \begin{cases}
39, & x \leq 200 \\
39 + 0.35(x - 200), & x > 200
\end{cases}
\][/tex]
- Option B is incorrect because it incorrectly places the conditions and does not make sense logically according to the problem's description.
- Option C is incorrect because it only accounts for the additional per-minute charge without adding the flat rate.
- Option D is incorrect because it misrepresents the additional charges beyond 200 minutes.
Thus, the correct answer is:
- A. \( f(x) = \begin{cases}
39, & x \leq 200 \\
39 + 0.35(x - 200), & x > 200
\end{cases} \)
