Answer :
To determine which piecewise function correctly represents the charges based on Tracy's cell phone plan, let's break down the information provided and match it to the corresponding function.
1. Flat Rate and Free Minutes:
- Tracy's plan includes 250 free minutes each month.
- The flat rate for these 250 minutes is \[tex]$29. 2. Charge for Minutes Over 250: - If Tracy uses more than 250 minutes, there is an additional charge of \$[/tex]0.35 per minute for every minute over the 250 minutes.
By analyzing the options and the given conditions, let's match them step by step.
### Option A
[tex]\[ f(x)=\left\{\begin{array}{c} 29, \, x \leq 250 \\ 29 + 0.35(x - 250), \, x > 250 \end{array}\right\} \][/tex]
- For [tex]\(x \leq 250\)[/tex]:
- The cost is \[tex]$29, which matches our free minutes flat rate. - For \(x > 250\): - The cost would be \$[/tex]29 for the first 250 minutes and an additional \[tex]$0.35 for each minute over 250. - This matches the form \( 29 + 0.35 (x - 250) \). ### Option B \[ f(x)=\left\{\begin{array}{c} 29, \, x \leq 250 \\ 0.35x, \, x > 250 \end{array}\right\} \] - For \(x \leq 250\): - The cost is \$[/tex]29, which is correct.
- For [tex]\(x > 250\)[/tex]:
- The cost would be [tex]\(0.35x\)[/tex] which does not account for the initial \[tex]$29 or ensure the correct charge for \(x > 250\) minutes. ### Option C \[ f(x)=\left\{\begin{array}{c} 29, \, x > 250 \\ 29 + 0.35x, \, x \leq 250 \end{array}\right\} \] - For \(x > 250\): - The cost given is \$[/tex]29, which is incorrect as it does not account for the additional minutes.
- For [tex]\(x \leq 250\)[/tex]:
- The cost given is [tex]\( 29 + 0.35x \)[/tex], which is incorrect because it charges extra for the first 250 minutes.
### Option D
[tex]\[ f(x)=\left\{\begin{array}{c} 29, \, x \leq 250 \\ 29 + 0.35x, \, x > 250 \end{array}\right\} \][/tex]
- For [tex]\(x \leq 250\)[/tex]:
- The cost is \$29, which is correct.
- For [tex]\(x > 250\)[/tex]:
- The cost would be [tex]\( 29 + 0.35x \)[/tex], which is incorrect because it does not account for only the minutes over 250.
Based on the evaluations, Option A correctly represents the charges based on Tracy's cell phone plan, as it addresses both the flat rate for 250 minutes and the additional charges correctly.
Thus, the correct piecewise function is:
[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 Option A.
1. Flat Rate and Free Minutes:
- Tracy's plan includes 250 free minutes each month.
- The flat rate for these 250 minutes is \[tex]$29. 2. Charge for Minutes Over 250: - If Tracy uses more than 250 minutes, there is an additional charge of \$[/tex]0.35 per minute for every minute over the 250 minutes.
By analyzing the options and the given conditions, let's match them step by step.
### Option A
[tex]\[ f(x)=\left\{\begin{array}{c} 29, \, x \leq 250 \\ 29 + 0.35(x - 250), \, x > 250 \end{array}\right\} \][/tex]
- For [tex]\(x \leq 250\)[/tex]:
- The cost is \[tex]$29, which matches our free minutes flat rate. - For \(x > 250\): - The cost would be \$[/tex]29 for the first 250 minutes and an additional \[tex]$0.35 for each minute over 250. - This matches the form \( 29 + 0.35 (x - 250) \). ### Option B \[ f(x)=\left\{\begin{array}{c} 29, \, x \leq 250 \\ 0.35x, \, x > 250 \end{array}\right\} \] - For \(x \leq 250\): - The cost is \$[/tex]29, which is correct.
- For [tex]\(x > 250\)[/tex]:
- The cost would be [tex]\(0.35x\)[/tex] which does not account for the initial \[tex]$29 or ensure the correct charge for \(x > 250\) minutes. ### Option C \[ f(x)=\left\{\begin{array}{c} 29, \, x > 250 \\ 29 + 0.35x, \, x \leq 250 \end{array}\right\} \] - For \(x > 250\): - The cost given is \$[/tex]29, which is incorrect as it does not account for the additional minutes.
- For [tex]\(x \leq 250\)[/tex]:
- The cost given is [tex]\( 29 + 0.35x \)[/tex], which is incorrect because it charges extra for the first 250 minutes.
### Option D
[tex]\[ f(x)=\left\{\begin{array}{c} 29, \, x \leq 250 \\ 29 + 0.35x, \, x > 250 \end{array}\right\} \][/tex]
- For [tex]\(x \leq 250\)[/tex]:
- The cost is \$29, which is correct.
- For [tex]\(x > 250\)[/tex]:
- The cost would be [tex]\( 29 + 0.35x \)[/tex], which is incorrect because it does not account for only the minutes over 250.
Based on the evaluations, Option A correctly represents the charges based on Tracy's cell phone plan, as it addresses both the flat rate for 250 minutes and the additional charges correctly.
Thus, the correct piecewise function is:
[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 Option A.
