1. **Problem Statement:** We want to maximize the total profit from producing cakes and pastries given constraints on baking and decoration hours. 2. **Define Variables:** Let $x$ = number of cakes produced per week. Let $y$ = number of pastries produced per week. 3. **Objective Function:** Maximize profit $P = 500x + 200y$ 4. **Constraints:** - Baking time: Each cake requires 3 hours, each pastry 1 hour, total baking hours available = 120. $$3x + y \leq 120$$ - Decoration time: Each cake requires 2 hours, each pastry 1 hour, total decoration hours available = 80. $$2x + y \leq 80$$ - Non-negativity: $$x \geq 0, \quad y \geq 0$$ 5. **Summary:** The linear programming model is: $$\text{Maximize } P = 500x + 200y$$ subject to $$3x + y \leq 120$$ $$2x + y \leq 80$$ $$x \geq 0, \quad y \geq 0$$ This model can be solved using graphical or simplex methods to find the values of $x$ and $y$ that maximize profit.