This appears to be a linear programming problem involving optimization for a café that offers two services: Coffee and Snacks. Let's break down the information provided to set up the problem:

1. Variables: Let $x$ represent the number of Coffee services and $y$ represent the number of Snacks services the café offers each week.

2. Constraints:

   • Barista A: Available for 40 hours. Coffee service takes 3 hours, and Snacks service takes 5 hours.
   • Barista B: Available for 30 hours. Coffee service takes 2 hours, and Snacks service takes 4 hours.
   • Weekly Maximum Budget: 120 services.

   This leads to the following system of inequalities based on the available hours:

   • $3x + 5y \leq 40$ (Barista A’s constraint)
   • $2x + 4y \leq 30$ (Barista B’s constraint)
   • $x + y \leq 120$ (service count constraint)

3. Objective Function: If the objective is to maximize profit, you would need a profit function, such as $P = px + qy$, where $p$ and $q$ are the profit contributions of Coffee and Snacks services respectively.

The next steps are:

• Graph these inequalities on a coordinate plane to identify the feasible region.
• Evaluate the objective function at each vertex of the feasible region to determine the optimal solution.

Let me know if you need the graph plotted or further explanation on setting up and solving the inequalities!

Here are five related questions for deeper understanding:

1. How do we find the feasible region for a linear programming problem?
2. What happens if a constraint changes, such as Barista B having only 20 hours available?
3. How can the café adjust if they want to maximize profit while also minimizing service time?
4. Why are the vertices of the feasible region key in determining the solution?
5. How does the profit per service impact the optimal solution?

Tip: In linear programming, always check each vertex of the feasible region to find the optimal value for maximization or minimization.