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.

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!

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.