The problem is a Linear Programming Maximization problem. Let's break it down step by step using Polya’s four-step problem-solving method.

I. Understanding the Problem

We need to maximize the profit from renting out two types of costumes: Fantasy and Historical. Here's what we know:

• Fantasy Costume Rental:

  • Requires 1 hour of assistance from Member A.
  • Requires 2 hours of assistance from Member B.
  • Profit per Fantasy rental is ₱80.

• Historical Costume Rental:

  • Requires 3 hours of assistance from Member A.
  • Requires 1 hour of assistance from Member B.
  • Profit per Historical rental is ₱100.

The total hours available are:

• Member A: 20 hours per week.
• Member B: 30 hours per week.

The goal is to determine how many Fantasy and Historical costumes should be rented each week to maximize profit.

II. Make a Plan

To solve this using linear programming, we'll define the decision variables, constraints, and the objective function:

1. Let x be the number of Fantasy costumes rented per week.
2. Let y be the number of Historical costumes rented per week.

Objective Function (to maximize profit): $Z = 80x + 100y$ where Z is the total profit.

Constraints (based on time available for Members A and B):

• Member A's time constraint: $1x + 3y \leq 20$ (since Member A has 20 hours available).
• Member B's time constraint: $2x + 1y \leq 30$ (since Member B has 30 hours available).
• Non-negativity constraints: $x \geq 0, y \geq 0$ (you can't rent a negative number of costumes).

III. Carry out the Plan

1. Solve the system of inequalities graphically or using a linear programming method such as the Simplex Method.
2. Identify the feasible region based on the constraints.
3. Evaluate the objective function at the vertices (corner points) of the feasible region to find the point that maximizes the profit.

IV. Look Back

Once the solution is found, check if it satisfies all the constraints (especially the time availability for both members) and if the profit is indeed maximized at that point.

Would you like me to proceed with solving this problem graphically or using a specific method?

Here are some related questions to extend understanding:

1. How can we solve the problem using the Simplex Method?
2. What happens if Member A or B’s availability changes?
3. What if the profit per costume changes?
4. Can you formulate a similar problem with different rental conditions?
5. How would we solve this problem without using linear programming?

Tip: When dealing with linear programming problems, always start by defining decision variables and constraints clearly!