1. **Stating the problem:** Alea Bakery makes two types of cakes: brownies and bika ambon. Brownies require 4 kg of wheat flour and 2 hours to make. Bika ambon requires 4 kg of wheat flour, 6 kg of tapioca flour, and 1 hour to make. Available resources are 120 kg wheat flour, 60 kg tapioca flour, and 40 hours of labor. The profit function to maximize is $$Z = 40x_1 + 30x_2$$ where $x_1$ is the number of brownies and $x_2$ is the number of bika ambon. 2. **Formulating constraints:** - Wheat flour: $$4x_1 + 4x_2 \leq 120$$ - Tapioca flour: $$6x_2 \leq 60$$ - Labor hours: $$2x_1 + x_2 \leq 40$$ - Non-negativity: $$x_1 \geq 0, x_2 \geq 0$$ 3. **Simplify constraints:** - Wheat flour: $$x_1 + x_2 \leq 30$$ - Tapioca flour: $$x_2 \leq 10$$ - Labor hours: $$2x_1 + x_2 \leq 40$$ 4. **Find feasible region vertices by solving intersections:** - Intersection of $$x_1 + x_2 = 30$$ and $$x_2 = 10$$: $$x_1 + 10 = 30 \Rightarrow x_1 = 20$$ - Intersection of $$x_1 + x_2 = 30$$ and $$2x_1 + x_2 = 40$$: Subtract first from second: $$2x_1 + x_2 - (x_1 + x_2) = 40 - 30 \Rightarrow x_1 = 10$$ Then $$x_2 = 30 - 10 = 20$$ - Intersection of $$x_2 = 10$$ and $$2x_1 + x_2 = 40$$: $$2x_1 + 10 = 40 \Rightarrow 2x_1 = 30 \Rightarrow x_1 = 15$$ 5. **Evaluate profit function at vertices:** - At (0,0): $$Z = 40(0) + 30(0) = 0$$ - At (0,10): $$Z = 40(0) + 30(10) = 300$$ - At (20,10): $$Z = 40(20) + 30(10) = 800 + 300 = 1100$$ - At (10,20): $$Z = 40(10) + 30(20) = 400 + 600 = 1000$$ - At (15,10): $$Z = 40(15) + 30(10) = 600 + 300 = 900$$ 6. **Conclusion:** The maximum profit is $$Z = 1100$$ at $$x_1 = 20$$ brownies and $$x_2 = 10$$ bika ambon.