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Let's analyze and solve each part of the problem.

### a. Determine the total charge under each plan for 120 minutes of daytime calls and 40 minutes of evening calls.

#### Plan A:
- Daytime calls: $0.45 per minute for 120 minutes
- Evening calls: $0.20 per minute for 40 minutes
\[
\text{Cost of Plan A} = (120 \times 0.45) + (40 \times 0.20)
\]
\[
\text{Cost of Plan A} = 54 + 8 = 62 \, \text{dollars}
\]

#### Plan B:
- Daytime calls: $0.55 per minute for 120 minutes
- Evening calls: $1.15 per minute for 40 minutes
\[
\text{Cost of Plan B} = (120 \times 0.55) + (40 \times 1.15)
\]
\[
\text{Cost of Plan B} = 66 + 46 = 112 \, \text{dollars}
\]

#### Plan C:
- Flat rate of $80 for 200 minutes total, no charge up to 200 minutes.
- Since 120 + 40 = 160 minutes is under 200 minutes, the total cost is simply $80.
\[
\text{Cost of Plan C} = 80 \, \text{dollars}
\]

**Summary of costs**:
- Cost for Plan A: $62
- Cost for Plan B: $112
- Cost for Plan C: $80

---

### b. Find the optimal range of call minutes for each plan (daytime calls only).

We compare the costs of Plan A and Plan B based on daytime calls, as Plan C is a flat rate.

#### Plan A:
Cost per minute for daytime calls is $0.45.

#### Plan B:
Cost per minute for daytime calls is $0.55.

To find the range of minutes where each plan is optimal, let's calculate the break-even point where both plans cost the same. 

We set up the equation for the cost of both plans:
\[
0.45x = 0.55x
\]
We can calculate the total range of optimal calls for Plan A, Plan B, and Plan C.

---

### c. Indifference point between plans A and B for both daytime and evening calls.

We solve for the percentage of daytime calls that make the cost of Plan A and Plan B equal when considering both daytime and evening calls.

I'll proceed to calculate all values.### a. The total charges under each plan for 120 minutes of daytime calls and 40 minutes of evening calls are:

- **Cost for Plan A:** $62.00
- **Cost for Plan B:** $112.00
- **Cost for Plan C:** $80.00

### b. Next, let's determine the range of daytime call minutes where each plan is optimal.

I'll now calculate the indifference points for optimal minutes between Plans A, B, and C, which will be based solely on daytime calls.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can help walk through the calculations manually or assist with a different approach. Let me know how you'd like to proceed!

Would you like to:
1. Calculate the break-even point manually for daytime minutes?
2. Compare other aspects of the plans?
3. Explore further clarifications of the charges?
4. Focus on solving part **c** manually?
5. Investigate a different scenario involving the plans?

Tip: Understanding how break-even points are calculated helps in comparing different plans efficiently!

A real estate agent is considering changing her landline phone plan. There are three plans to choose from, each with a different cost structure based on the usage of daytime and evening calls. Determine the total charges under each plan, the optimal range of minutes for each plan, and the point of indifference between plans A and B.

Learn how to determine the most cost-effective phone plan for a real estate agent by calculating total charges, finding the optimal range of call minutes, and identifying indifference points between plans A and B. This step-by-step guide covers essential cost calculation and optimization techniques, suitable for high school students.

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