Ben has a cell phone plan that provides 200 free minutes each month for a flat rate of $39. For any minutes over 200, Ben is charged $0.35 per minute. Which of the following piecewise functions accurately represents charges based on Ben's cell phone plan?
A. [tex]f(x)=\left\{\begin{array}{c}39, x \leq 200 \\ 39+.35(x-200), x\ \textgreater \ 200\end{array}\right\}[/tex]
B. [tex]f(x)=\left\{\begin{array}{c}39, x \leq 200 \\ .35(x-200), x\ \textgreater \ 200\end{array}\right\}[/tex]
C. [tex]f(x)=\left\{\begin{array}{c}39, x \leq 200 \\ .35 x, x\ \textgreater \ 200\end{array}\right\}[/tex]
D. [tex]f(x)=\left\{\begin{array}{c}39, x\ \textgreater \ 200 \\ 39+.35, x \leq 200\end{array}\right\}[/tex]
Answer (2)
The cell phone plan costs $39 for up to 200 minutes.
For minutes over 200, the charge is $39 + $0.35 per extra minute.
The piecewise function is constructed based on these two conditions.
The correct piecewise function is A .
Explanation
Understanding the Cell Phone Plan Let's analyze Ben's cell phone plan to determine the correct piecewise function. The plan has a flat rate of $39 for the first 200 minutes. If Ben uses more than 200 minutes, he pays an additional $0.35 for each extra minute. We need to construct a piecewise function, f(x), that represents the total charge based on the number of minutes, x, used.
Cost for 200 Minutes or Less If Ben uses 200 minutes or less ( x ≤ 200 ), the cost is a flat $39. This is represented as:
f ( x ) = 39 , for x ≤ 200
Cost for More Than 200 Minutes If Ben uses more than 200 minutes ( 200"> x > 200 ), the cost is the flat rate of $39 plus $0.35 for each minute exceeding 200. The number of minutes exceeding 200 is ( x − 200 ) . Therefore, the cost for 200"> x > 200 is:
f ( x ) = 39 + 0.35 ( x − 200 ) , for 200"> x > 200
Constructing the Piecewise Function Combining these two conditions, the piecewise function is:
200 \end{cases}"> f ( x ) = { 39 , 39 + 0.35 ( x − 200 ) , x ≤ 200 x > 200
Identifying the Correct Option Now, we compare our constructed piecewise function with the given options:
A. 200\end{array}\right\}"> f ( x ) = { 39 , x ≤ 200 39 + .35 ( x − 200 ) , x > 200 } B. 200\end{array}\right\}"> f ( x ) = { 39 , x ≤ 200 .35 ( x − 200 ) , x > 200 } C. 200\end{array}\right\}"> f ( x ) = { 39 , x ≤ 200 .35 x , x > 200 } D. 200 \\ 39+.35, x \leq 200\end{array}\right\}"> f ( x ) = { 39 , x > 200 39 + .35 , x ≤ 200 }
Option A matches our constructed piecewise function.
Final Answer Therefore, the correct piecewise function that represents Ben's cell phone plan charges is option A.
A
Examples
Piecewise functions are useful in real life for modeling situations where different rules or conditions apply over different intervals. For example, consider a parking garage that charges a flat fee for the first two hours and then an hourly rate after that. A piecewise function can accurately represent the total parking cost based on the number of hours parked. Similarly, tax brackets, shipping costs, and tiered pricing models can all be effectively modeled using piecewise functions, providing clarity and accuracy in representing varying conditions.
The correct piecewise function representing Ben's cell phone plan is option A, which states that for 200 minutes or less, the cost is $39, and for minutes over 200, the cost is $39 plus $0.35 for each additional minute. This can be expressed as: f(x) = {39, if x ≤ 200; 39 + 0.35(x - 200), if x > 200}.
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