1. **State the problem:** We want to determine how many units of products A and B should be produced and sold to maximize total profit, given constraints on raw materials and labor hours. 2. **Define variables:** Let $x$ = units of product A produced and sold. Let $y$ = units of product B produced and sold. 3. **Constraints:** - Raw materials: Each unit of A requires 2 kg, each unit of B requires 3 kg, total available = 60 kg. $$2x + 3y \leq 60$$ - Labor hours: Each unit of A requires 4 hours, each unit of B requires 3 hours, total available = 96 hours. $$4x + 3y \leq 96$$ - Non-negativity: $$x \geq 0, \quad y \geq 0$$ 4. **Profit per unit:** - Profit from A = 40 per unit - Profit from B = 35 per unit 5. **Objective function:** Maximize total profit: $$P = 40x + 35y$$ 6. **Solve the linear programming problem:** Check corner points of the feasible region defined by constraints. - Intersection of constraints: Solve system: $$2x + 3y = 60$$ $$4x + 3y = 96$$ Subtract first from second: $$4x + 3y - (2x + 3y) = 96 - 60 \Rightarrow 2x = 36 \Rightarrow x = 18$$ Substitute $x=18$ into first: $$2(18) + 3y = 60 \Rightarrow 36 + 3y = 60 \Rightarrow 3y = 24 \Rightarrow y = 8$$ - Other corner points: At $x=0$: $$2(0) + 3y \leq 60 \Rightarrow y \leq 20$$ $$4(0) + 3y \leq 96 \Rightarrow y \leq 32$$ So $y=20$ at $x=0$. At $y=0$: $$2x + 3(0) \leq 60 \Rightarrow x \leq 30$$ $$4x + 3(0) \leq 96 \Rightarrow x \leq 24$$ So $x=24$ at $y=0$. 7. **Evaluate profit at corner points:** - At $(0,20)$: $$P = 40(0) + 35(20) = 700$$ - At $(24,0)$: $$P = 40(24) + 35(0) = 960$$ - At $(18,8)$: $$P = 40(18) + 35(8) = 720 + 280 = 1000$$ 8. **Conclusion:** Maximum profit is 1000 when producing and selling 18 units of A and 8 units of B. 9. **Regarding the revenue and cost functions:** Given $R = -3x^2 + 200x$ and $C = 2x^2 - 150x + 5000$ for a single product with $x$ in thousands, these seem unrelated to the two-product problem above or may represent a different scenario. **Final answer:** Produce and sell 18 units of product A and 8 units of product B to maximize profit. Maximum profit = 1000 (units of currency).