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The problem provided gives a scenario where a company is conducting surveys over the phone and needs to determine the optimal way to allocate resources (interviewers) based on the time it takes for each interviewer to complete their tasks.

The key function provided is:

\[
f(k) = 8(1 + k)^{-0.2}
\]

This function represents the time (in minutes) taken by an interviewer to make the \( k \)-th call, where \( k \) is the number of previous calls made by that interviewer. The company needs to complete 200 surveys and is considering different options to minimize costs based on the number of interviewers.

Let's analyze each option:

1. **Option a**: Hiring a single person to complete all 200 surveys.
2. **Option b**: Hiring one person for 150 surveys and another for 50 surveys.
3. **Option c**: Hiring two people, each to complete 100 surveys.
4. **Option d**: Hiring four people, each to complete 50 surveys.

To determine which option is the most efficient, we will need to calculate the total time taken for each option. The total time for any number of surveys is the sum of the times for each call (calculated using the given function).

### Let's start calculating step by step:

1. **Option a**: Calculate the total time for one person to make 200 calls.
2. **Option b**: Calculate the total time for one person to make 150 calls and another person to make 50 calls.
3. **Option c**: Calculate the total time for two people to make 100 calls each.
4. **Option d**: Calculate the total time for four people to make 50 calls each.

I'll calculate the values now.Here are the total times calculated for each option:

- **Option a**: One person makes all 200 calls → **688.66 minutes**.
- **Option b**: One person makes 150 calls, another person makes 50 calls → **770.85 minutes**.
- **Option c**: Two people, each making 100 calls → **787.66 minutes**.
- **Option d**: Four people, each making 50 calls → **898.44 minutes**.

### Conclusion:
The best option for the company, based on minimizing the total time spent on completing 200 surveys, is **Option a: Hiring a single person**. This option results in the least total time, **688.66 minutes**.

Let me know if you'd like further details or have any other questions.

Here are 5 related questions to expand the discussion:
1. What would the total time be if 250 surveys were needed instead of 200?
2. How does the cost differ if each interviewer charges based on the time spent?
3. How would the efficiency change if the power of the decay factor in the function was altered?
4. How would hiring 3 people for 67 surveys each compare to the current options?
5. What if each interviewer took a break after every 50 calls—how would that affect the calculation?

**Tip:** When calculating totals for repetitive tasks with varying efficiencies, using summation formulas like this can provide insights into resource allocation strategies.

Una compañía encargada de realizar encuestas por medio telefónico ha realizado estudios sobre el tiempo que toma un encuestador al realizar las encuestas (o llamadas) y determinó que este depende de la experiencia que adquiere en las llamadas que ha realizado anteriormente. La función que determina el tiempo en minutos utilizados por el encuestador hasta la llamada k, está dada por f(k) = 8(1 + k)^(-0.2). Si se debe pagar por minuto el trabajo para realizar 200 encuestas ¿Cuál de las siguientes opciones, representa la mejor para la empresa? a) Contratar una sola persona. b) Contratar una para 150 encuestas y otra para 50 encuestas. c) Contratar dos personas cada una 100 encuestas. d) Contratar cuatro personas cada una con 50 encuestas.

Learn how to optimize interviewer allocation in a company conducting surveys over the phone. This problem involves a decay function f(k) = 8(1 + k)^(-0.2) to calculate the time taken for calls based on experience. Explore how different hiring strategies impact the total time for 200 surveys.

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