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Let's go through this Linear Programming Application problem step by step.

### Problem Recap:
- A professor gives **objective** and **recall** quizzes.
- **Objective quiz**:
  - Student prep time: **15 minutes**
  - Average score: **7**
  - Grading time: **2.5 minutes**
- **Recall quiz**:
  - Student prep time: **30 minutes**
  - Average score: **5**
  - Grading time: **1.5 minutes**

### Objective:
Minimize the professor’s grading time while ensuring students spend at least **4.5 hours** (270 minutes) preparing and score at least **85 points**.

Define:
- \( x \): Number of **objective quizzes**.
- \( y \): Number of **recall quizzes**.

The objective function is the professor's total grading time:
\[ z = 2.5x + 1.5y \]

Now, let's break down the constraints:

### Constraints:

1. **Time Constraint** (Preparation):
   - \( 15x + 30y \geq 270 \) (At least 270 minutes of preparation time)
   
2. **Score Constraint** (Total Score):
   - \( 7x + 5y \geq 85 \) (At least 85 total points from quizzes)

3. **Quiz Requirement** (Number of quizzes):
   - \( x + y \geq 16 \) (At least 16 quizzes)

### Solving:

#### (a) Objective Function:
The objective function is already given as:
\[ z = 2.5x + 1.5y \]

#### (b) Graph the Feasible Region:
For this, you need to graph the three constraints:
1. \( 15x + 30y \geq 270 \) simplifies to \( x + 2y \geq 18 \).
2. \( 7x + 5y \geq 85 \) remains the same.
3. \( x + y \geq 16 \).

Plot these inequalities on a graph and find the intersection points, which will give you the **corner points** of the feasible region.

#### (c) Test the Corner Points:
Once you have the corner points of the feasible region, plug them into the objective function \( z = 2.5x + 1.5y \) to find which one minimizes the grading time.

- The **winning corner point** will be the one that results in the smallest value of \( z \).
- The corresponding value of \( z \) will be the **optimum value** of the objective function.

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If you'd like to see detailed steps for solving, let me know. You can also upload a graph of the feasible region if you've drawn it!

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### Questions for Deeper Understanding:
1. How can we interpret the results of the optimum value in a real classroom setting?
2. What happens if the professor wants to increase the minimum number of quizzes to 20?
3. How would the problem change if the preparation time for a recall quiz was reduced to 20 minutes?
4. How does changing the professor’s grading time for each quiz affect the objective function?
5. What could be a real-world scenario where minimizing grading time is not the primary objective?

### Tip:
In linear programming, always check if your corner points satisfy **all** constraints before selecting the optimal solution.

A professor gives two types of quizzes, objective and recall. He is planning to give at least 16 quizzes this quarter. The student preparation time for an objective quiz is 15 minutes and for a recall quiz 30 minutes. The professor would like a student to spend at least 4.5 hours (270 minutes) preparing for these quizzes above and beyond the normal study time. The professor would like the students to score at least 85 points on all quizzes. It takes the professor 2.5 minutes to grade an objective quiz, and 1.5 minutes to grade a recall type quiz. How many of each type should he give in order to minimize his grading time?

Solve a linear programming problem where a professor seeks to minimize grading time for quizzes while ensuring students meet preparation and score requirements. Explore inequalities, objective functions, and optimization techniques. Suitable for Grades 11-12 or college students.

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