Problem 5 6a: Real Estate Agent Considering Change 690405
Problem 5 6a Real Estate Agent Is Considering Changing Her Cell Phone
A real estate agent is evaluating three different cell phone plans, each with a fixed monthly service charge of $20. The plans differ in their per-minute charges for daytime and evening calls, and Plan C offers a flat rate with a usage limit before additional charges apply. The goal is to determine the total costs for specific usage scenarios, identify the call ranges where each plan is most economical, and analyze the point of indifference between plans based on call distribution.
Paper For Above instruction
Cell phone plans are a critical consideration for professionals like real estate agents, who often need reliable and cost-effective communication solutions. Analyzing different plans involves understanding their cost structure relative to expected call usage patterns. The three plans under consideration include varied per-minute rates and a flat-rate option with overage charges, necessitating a detailed comparison to identify the most economical choice for specific and general usage scenarios.
Cost Calculation for Specific Call Usage
Given usage of 140 minutes during the day and 60 minutes in the evening, the total number of call minutes per month sums to 200 minutes. Calculating the total monthly cost for each plan involves accounting for the fixed monthly fee and the costs associated with the per-minute call rates.
Plan A
Plan A charges $0.39 per minute for daytime calls and $0.19 per minute for evening calls, with a fixed fee of $20. The total cost is computed as:
- Daytime: 140 minutes × $0.39 = $54.60
- Evening: 60 minutes × $0.19 = $11.40
Total variable cost: $54.60 + $11.40 = $66.00
Total cost including fixed fee: $20 + $66.00 = $86.00
Cost for Plan A: 86.00
Plan B
Plan B charges $0.49 per minute for daytime and $0.14 per minute for evening calls, with a fixed fee of $20. The total cost is:
- Daytime: 140 × $0.49 = $68.60
- Evening: 60 × $0.14 = $8.40
Total variable cost: $68.60 + $8.40 = $77.00
Total cost including fixed fee: $20 + $77.00 = $97.00
Cost for Plan B: 97.00
Plan C
Plan C features a flat rate of $75 for up to 225 minutes per month; additional minutes are charged at $0.36 per minute. Since total usage is 200 minutes, which is within the included 225 minutes, the cost is simply:
Flat rate: $75
Cost for Plan C: 75.00
Comparison and Optimal Plan Range for Daytime Calls
To determine the ranges where each plan is most cost-effective when primarily considering daytime calls, we must compare the total costs as functions of call minutes.
Plan A vs. Plan B
Set the costs equal to find the break-even point:
$0.39 \times 140 + $0.19 \times 60 + 20 \text{ (fixed)} = 0.49 \times 140 + 0.14 \times 60 + 20$
Simplify:
$54.60 + 11.40 + 20 = 68.60 + 8.40 + 20$
Both sides equal $86.00 and $97.00 respectively, which aligns with earlier calculations. To find the call minutes where Plan A becomes more economical than Plan B, express total cost for each plan as a function of daytime minutes, \(x\):
- For Plan A:
\[
\text{Cost}_A = 20 + 0.39x + 0.19(200 - x) = 20 + 0.39x + 38 - 0.19x = 58 + 0.20x
\]
- For Plan B:
\[
\text{Cost}_B = 20 + 0.49x + 0.14(200 - x) = 20 + 0.49x + 28 - 0.14x = 48 + 0.35x
\]
Set \(\text{Cost}_A = \text{Cost}_B\):
\[
58 + 0.20x = 48 + 0.35x
\]
\[
58 - 48 = 0.35x - 0.20x
\]
\[
10 = 0.15x
\]
\[
x = \frac{10}{0.15} \approx 66.67
\]
Thus, Plan A is more economical when daytime calls are up to approximately 67 minutes. For call durations above this point, Plan B becomes more cost-effective.
Range of optimality for plans:
- Plan A is optimal from 0 to approximately 66 minutes of daytime calls.
- Plan B is optimal from approximately 67 minutes onward.
Since the original usage data involve 140 minutes of daytime calls, in this specific case, Plan B is the most economical.
---
Indifference Point Between Plans A and B Based on Call Distribution
To find the percentage of total call minutes that would make the agent indifferent between plans A and B, we revisit the cost functions considering variable call distributions. Let \(x\) be the number of daytime call minutes, and total call minutes are \(T\).
- Cost for Plan A:
\[
\text{Cost}_A = 20 + 0.39x + 0.19(T - x) = 20 + 0.39x + 0.19T - 0.19x = 20 + 0.19T + 0.20x
\]
- Cost for Plan B:
\[
\text{Cost}_B = 20 + 0.49x + 0.14(T - x) = 20 + 0.49x + 0.14T - 0.14x = 20 + 0.14T + 0.35x
\]
Set \(\text{Cost}_A = \text{Cost}_B\):
\[
20 + 0.19T + 0.20x = 20 + 0.14T + 0.35x
\]
Subtract 20 from both sides:
\[
0.19T + 0.20x = 0.14T + 0.35x
\]
Bring like terms together:
\[
0.19T - 0.14T = 0.35x - 0.20x
\]
\[
0.05T = 0.15x
\]
\[
x = \frac{0.05}{0.15} T = \frac{1}{3} T
\]
Expressed as a percentage:
\[
\frac{x}{T} \times 100 = \frac{1}{3} \times 100 \approx 33.33\%
\]
Point of indifference:
The agent would be indifferent between plans A and B when approximately 33.33% of her total call minutes are daytime calls.
This insight indicates that if more than a third of total call duration occurs during daytime, Plan B is typically more economical, whereas with a lesser proportion, Plan A provides savings.
---
Conclusion
Analyzing cell phone plans in the context of estimated call usage involves carefully considering fixed and variable costs. For the given scenario, Plan C is most economical for modest usage, with a flat fee of $75 covering up to 225 minutes. When usage exceeds this threshold, additional charges apply, making Plans A and B more suitable depending on the distribution between daytime and evening calls. Our calculations show that Plan A is preferable for minimal daytime calls, but as the daytime call duration increases, Plan B becomes more cost-effective. The critical point, at approximately 33.33% of total call minutes dedicated to daytime calls, determines the agent's optimal plan choice based on usage patterns.
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