A Real Estate Agent Is Considering Changing Her Cell Phone P

A Real Estate Agent Is Considering Changing Her Cell Phone Plan There

A real estate agent is evaluating three different cell phone plans to determine which one offers the best value based on expected call usage. The plans have specific costs associated with daytime and evening calls, as well as fixed monthly charges and usage limits in some cases. The goal is to compare the total monthly costs for a specific call pattern, identify the range of call minutes where each plan is most economical, and determine the call distribution at which the agent would be indifferent between two plans.

Problem parts:

1. Calculate the total monthly charges for each plan based on a usage of 116 minutes of daytime calls and 40 minutes of evening calls.
2. Identify the range of total call minutes where each plan is the most economical when the agent exclusively makes daytime calls.
3. Determine the percentage of daytime call minutes at which the agent would be equally satisfied with either Plan A or Plan B.

Calculations

Part 1: Total charges with specific usage

Given call usage: 116 minutes daytime, 40 minutes evening.

Plan A:

• Monthly service charge: $29
• Daytime call rate: $0.45 per minute
• Evening call rate: $0.29 per minute

Total cost for Plan A:

= 29 + (116 × 0.45) + (40 × 0.29)

= 29 + 52.2 + 11.6

= 92.8

Rounded to the nearest whole number: 93

Plan B:

• Monthly service charge: $29
• Daytime call rate: $0.51 per minute
• Evening call rate: $0.17 per minute

Total cost for Plan B:

= 29 + (116 × 0.51) + (40 × 0.17)

= 29 + 59.16 + 6.8

= 95.96

Rounded: 96

Plan C:

• Monthly fee: $95
• Includes 253 minutes; charges beyond that are $0.40 per minute

Total minutes used: 116 + 40 = 156.

Since 156 > 253, the overage minutes:

= 156 - 253 = -97

But since the total usage exceeds the plan limit, the charge for overage is:

= 95 + (156 - 253) × 0.40 = 95 + (-97) × 0.40

As minutes exceeding the cap are positive, adjust the calculation accordingly:

= 95 + (156 - 253) × 0.40

= 95 + 0. (Note: Since total usage exceeds limit, overage = 156 - 253 = -97, which is negative--incorrect, so revisit.)

Actually, total minutes are 156, and plan includes 253 minutes, so no overage charge needed.

Total cost for Plan C:

= 95 (since usage is below 253 minutes)

Rounded: 95

Summary of costs:

• Plan A: 93
• Plan B: 96
• Plan C: 95

Part 2: Range of call minutes for which each plan is optimal

We analyze when each plan is cheaper based on total call minutes (Daytime + Evening).

Assuming only daytime calls for simplicity, the total costs are as follows:

Plan A cost: 29 + 0.45 * total minutes

Plan B cost: 29 + 0.51 * total minutes

Plan C cost: 95 if total minutes ≤ 253; otherwise, 95 + 0.40 * (total minutes - 253)

To find when Plan A is optimal over Plan B, set their costs equal:

29 + 0.45 x = 29 + 0.51 x

Simplify:

0.45x = 0.51x

=> 0 = 0.06x

=> x = 0

This indicates Plan A is cheaper for all call minutes less than where costs intersect. Let's find where Plan A becomes more costly than Plan B for larger x:

At intersection:

0.45x = 0.51x

=> no meaningful intersection except at zero. So, Plan A is always cheaper than Plan B for small call volumes, particularly less than the critical point where costs tie.

Similarly, compare Plans A and C:

Since Plan C is flat at $95 for 253 minutes or fewer, Plan A is cheaper when:

29 + 0.45x

=> 0.45x

=> x

Beyond 146.67 minutes, Plan C may become more economical, especially since Plan C's overage rate is lower than Plan B's.

Similarly, for Plan B:

0.51x

=> x

Summary:

- Plan A is optimal for total call minutes less than approximately 147.

- Plan C becomes economical when usage exceeds approximately 147 minutes.

- Plan B is least favorable at lower minutes but becomes competitive depending on overage costs.

Part 3: Indifference point between Plans A and B based on call minute distribution

We seek the percentage of daytime minutes at which costs of Plans A and B are equal, given total call minutes (x). The total costs are:

Plan A: 29 + 0.45 (percentage of daytime minutes) x + 0.29 * (remaining minutes)

Plan B: 29 + 0.51 (percentage of daytime minutes) x + 0.17 * (remaining minutes)

The difference:

(0.45 p x + 29) = (0.51 p x + 29)

Simplify:

0.45p x = 0.51p x

=> 0 = 0.06 p x

=> p = 0

Since the costs are based on per-minute rates and fixed charges, the indifference point occurs where:

0.45 p x + 29 = 0.51 p x + 29

=> 0.45 p x = 0.51 p x

=> again, only at p = 0 or x = 0, which isn't meaningful.

Instead, focus on the case when total minutes are identical, and solve for p:

Set:

29 + 0.45 p x + 0.29 (remaining minutes) = 29 + 0.51 p x + 0.17 (remaining minutes)

Assuming total minutes x, the total minutes allocated to daytime:

Total cost difference:

(0.45 p + 0.29 (1 - p)) x + 29 = (0.51 p + 0.17 (1 - p)) x + 29

Subtract 29:

(0.45 p + 0.29 - 0.29 p) x = (0.51 p + 0.17 - 0.17 p) x

Simplify:

(0.45 p - 0.29 p + 0.29) x = (0.51 p - 0.17 p + 0.17) x

(0.16 p + 0.29) x = (0.34 p + 0.17) x

Divide both sides by x:

0.16 p + 0.29 = 0.34 p + 0.17

Solve for p:

0.29 - 0.17 = 0.34 p - 0.16 p

0.12 = 0.18 p

p = 0.12 / 0.18 ≈ 0.6667

Expressed as a percentage: 66.67%

Rounded to the nearest whole percent: 67%

Final conclusions

• Cost for Plan A: 93 dollars
• Cost for Plan B: 96 dollars
• Cost for Plan C: 95 dollars

Plan A is optimal for call minutes less than approximately 147. Plan C becomes preferable beyond that point. Plan B generally is less economical in the contexts considered. The agent would be indifferent between Plans A and B at approximately 67% of call minutes being daytime calls, assuming total call minutes are sufficiently large to render fixed costs negligible.

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