1. **Problem Statement:** We want to determine the number of units of products A ($X_1$) and B ($X_2$) to produce weekly to maximize profit. 2. **Define Variables:** Let $X_1$ = number of units of product A Let $X_2$ = number of units of product B 3. **Constraints:** - Machine time: Each unit of A takes 10 minutes, each unit of B takes 20 minutes. Total machine time available = 35 hours = $35 \times 60 = 2100$ minutes. So, $$10X_1 + 20X_2 \leq 2100$$ - Raw material: A requires 1 kg per unit, B requires 0.5 kg per unit. Total raw material available = 600 kg. So, $$X_1 + 0.5X_2 \leq 600$$ - Market constraint on B: minimum 800 units. So, $$X_2 \geq 800$$ - Non-negativity: $$X_1 \geq 0, \quad X_2 \geq 0$$ 4. **Profit Function:** Profit per unit of A = selling price - cost = $10 - 5 = 5$ Profit per unit of B = $8 - 6 = 2$ Total profit to maximize: $$P = 5X_1 + 2X_2$$ 5. **Formulate the Linear Programming Problem:** Maximize $$P = 5X_1 + 2X_2$$ subject to $$10X_1 + 20X_2 \leq 2100$$ $$X_1 + 0.5X_2 \leq 600$$ $$X_2 \geq 800$$ $$X_1, X_2 \geq 0$$ 6. **Solve Constraints:** Since $X_2 \geq 800$, substitute $X_2 = 800$ to check feasibility. - Machine time: $$10X_1 + 20 \times 800 \leq 2100 \Rightarrow 10X_1 + 16000 \leq 2100$$ This is impossible since $16000 > 2100$, so $X_2=800$ alone violates machine time. Therefore, no feasible solution with $X_2=800$ unless $X_1$ is negative, which is not allowed. 7. **Check if constraints are consistent:** Since $X_2 \geq 800$ and machine time is limited, the problem is infeasible as stated. 8. **Re-examine machine time constraint:** Maximum machine time is 2100 minutes. If $X_2=800$, machine time used by B alone is $20 \times 800 = 16000$ minutes, which exceeds 2100. This suggests a possible error in the problem statement or units. 9. **Assuming machine time is 35 hours per day or 35 hours per week is correct, then 2100 minutes is correct.** Since $20X_2 \leq 2100$ and $X_2 \geq 800$, no solution exists. 10. **Conclusion:** The constraints are contradictory; no feasible production plan satisfies all constraints. **Final answer:** No feasible solution exists under the given constraints to maximize profit.