To solve this Linear Programming (LP) problem, we’ll need to:

1. Define the variables.
2. Set up the objective function to maximize.
3. Write the constraints based on the hours available for each barista.

Step 1: Define the Variables

Let:

• $x$ = number of Coffee services provided each week.
• $y$ = number of Snacks services provided each week.

Step 2: Objective Function

The goal is to maximize the weekly profit.

Profit per Coffee service = ₱80
Profit per Snacks service = ₱120

The objective function, representing total profit $P$, is: $P = 80x + 120y$

Step 3: Constraints

The cafe has limited hours for each barista, leading to constraints based on their weekly availability.

Barista A's Constraint

Each Coffee service requires 1 hour of Barista A’s time, and each Snacks service requires 3 hours. Since Barista A is available for a maximum of 25 hours weekly: $1x + 3y \leq 25$

Barista B's Constraint

Each Coffee service requires 2 hours of Barista B’s time, and each Snacks service requires 1 hour. Since Barista B is available for a maximum of 30 hours weekly: $2x + 1y \leq 30$

Non-negativity Constraints

The number of services cannot be negative: $x \geq 0 \quad \text{and} \quad y \geq 0$

Step 4: Solve the System

The problem now is: Maximize: $P = 80x + 120y$ Subject to: $1x + 3y \leq 25$ $2x + 1y \leq 30$ $x \geq 0, \quad y \geq 0$

Step 5: Graphical Solution

To find the optimal solution, we can solve this graphically by plotting each constraint and finding the feasible region. We then evaluate the profit function $P = 80x + 120y$ at each vertex of the feasible region to determine which combination of $x$ and $y$ maximizes $P$.

Let's go over these calculations to get the exact answer. If you'd like further details on solving it graphically or using another method, just let me know!

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Would you like a step-by-step calculation for each constraint? Here are related questions to explore further:

1. How would you solve a linear programming problem with more than two variables?
2. What is the importance of defining non-negativity constraints in LP problems?
3. How can duality theory be applied to this maximization problem?
4. What methods other than graphical analysis can solve this LP problem?
5. How does changing the profit per service impact the optimal solution?

Tip: Always check each constraint to ensure that the chosen values for $x$ and $y$ satisfy the feasible region for the problem.