To solve this problem, we need to formulate a linear programming model to find the optimal product mix (the number of products X and Y to produce) that maximizes profit while satisfying the machine capacity constraints.

Step 1: Define the Decision Variables

Let:

• $x$ = number of units of Product X to produce
• $y$ = number of units of Product Y to produce

Step 2: Define the Objective Function

The objective is to maximize profit, where the profit contributions are:

• ₱10 for each unit of Product X
• ₱20 for each unit of Product Y

Thus, the objective function (profit) is: $\text{Maximize} \ Z = 10x + 20y$

Step 3: Define the Constraints

The available capacities for the machines are:

• Machine 1: 30 hours
• Machine 2: 36 hours
• Machine 3: 20 hours

We need to express these constraints in terms of $x$ and $y$.

1. Machine 1 constraint: Product X requires 5 hours, and Product Y requires 3 hours on Machine 1. The total time on Machine 1 should be less than or equal to the available capacity of 30 hours: $5x + 3y \leq 30$

2. Machine 2 constraint: Product X requires 3 hours, and Product Y requires 6 hours on Machine 2. The total time on Machine 2 should be less than or equal to the available capacity of 36 hours: $3x + 6y \leq 36$

3. Machine 3 constraint: Product X requires 2 hours, and Product Y requires 5 hours on Machine 3. The total time on Machine 3 should be less than or equal to the available capacity of 20 hours: $2x + 5y \leq 20$

4. Non-negativity constraints: Since we can't produce a negative quantity of products, we also have: $x \geq 0, \quad y \geq 0$

Step 4: Set up the Linear Programming Problem

We want to maximize the profit $Z = 10x + 20y$ subject to the following constraints:

5x + 3y &\leq 30 \\ 3x + 6y &\leq 36 \\ 2x + 5y &\leq 20 \\ x &\geq 0 \\ y &\geq 0 \end{aligned}$$ ### Step 5: Solve the Linear Programming Problem Now, let’s solve the problem. The graphical or simplex method can be used to find the optimal solution. Would you like me to continue and solve it step-by-step or do you prefer a quick result using an optimization method?