The budget for a project on voting trends includes ​$38003800 for hiring undergraduate​ students, graduate​ students, and faculty members to conduct interviews on the day before an election. Each undergraduate student will conduct 2828 interviews for​ $100. Each graduate student will conduct 3232 interviews for​ $150. Each faculty member will conduct 3434 interviews for​ $200. No more than 2121 interviewers can be hired. How many of each type of interviewer should be hired in order to maximize the number of​ interviews? What is the maximum number of​ interviews?
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Part 1
enter your response here undergraduate​ students, enter your response here graduate​ students, and enter your response here faculty members should be hired to maximize the number of interviews.
Part 2
The maximum number of interviews is enter your response here.
​(Type a whole​ number.)

This problem can be solved using linear programming to maximize the number of interviews while staying within budget and hiring constraints. Let's define the variables:

- Let $ x $ be the number of undergraduate students hired.
- Let $ y $ be the number of graduate students hired.
- Let $ z $ be the number of faculty members hired.

### Objective Function
The goal is to maximize the number of interviews:

\[
\text{Maximize } I = 28x + 32y + 34z
\]

### Constraints
1. The budget constraint:
   \[
   100x + 150y + 200z \leq 38003800
   \]
   
2. The hiring constraint (no more than 2121 interviewers):
   \[
   x + y + z \leq 2121
   \]

3. Non-negativity constraints:
   \[
   x \geq 0, \quad y \geq 0, \quad z \geq 0
   \]

Given the large budget number, I assume there may have been a typo in the problem, so I will assume a budget of $380,000 instead. Let's solve this problem using the smaller budget to avoid impractical results. If you'd like to proceed with the larger number, please let me know.

### Revised Budget Constraints
\[
100x + 150y + 200z \leq 380000
\]

### Solving the Problem
Now, I will solve the linear programming problem to determine the optimal number of undergraduate students, graduate students, and faculty members to hire.

Let's proceed with solving it mathematically.To maximize the number of interviews while staying within the budget and hiring constraints:

- **0** undergraduate students should be hired,
- **884** graduate students should be hired,
- **1237** faculty members should be hired.

### Maximum Number of Interviews
The maximum number of interviews that can be conducted is **70,346**.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How would the result change if the budget were increased?
2. What if only a specific number of undergraduate students could be hired?
3. How does changing the number of interviews each person can conduct affect the outcome?
4. What if the cost per interviewer type was different?
5. How could constraints on individual groups (like max number of faculty) alter the solution?

**Tip:** When solving problems with large numbers, verifying any potential typographical errors in the input can ensure the solution is more practical.

The budget for a project on voting trends includes $38003800 for hiring undergraduate students, graduate students, and faculty members to conduct interviews on the day before an election. Each undergraduate student will conduct 28 interviews for $100. Each graduate student will conduct 32 interviews for $150. Each faculty member will conduct 34 interviews for $200. No more than 2121 interviewers can be hired. How many of each type of interviewer should be hired in order to maximize the number of interviews? What is the maximum number of interviews?

This problem explores how to maximize the number of interviews conducted by undergraduate, graduate students, and faculty members under a $380,000 budget. The solution involves solving a linear programming problem and understanding key constraints such as budget limits and the number of interviewers. A practical example of optimization techniques in resource allocation.

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