Question: User: Question 3 \n\n\n\n\n\n\n\n\n\n[14 marks]\nAn electronics manufacturer produces two types of components: Circuit Boards anMicrocontrollers. Each Circuit Board requires 3 hours of soldering time and 2 hours of qualitinspection. Each Microcontroller requires 4 hours of soldering time and 1 hour of qualitinspection. The factory has a maximum of 120 hours available for soldering and 50 hours forinspection per week. Warehouse space limits the total combined output to no more than 35units per week. The profit per Circuit Board is N$20, and the profit per Microcontroller isN$30.\n\n3.1 Formulate a linear programming model. [5]3.2 Determine how many units of each component should be produced to maximize totalweekly profit. [6\n3.3 Calculate the maximum total weekly profit. [3
1. **Problem Statement:**\nWe have two products: Circuit Boards (let $x$ be the number of Circuit Boards) and Microcontrollers (let $y$ be the number of Microcontrollers).\n\n2. **Formulate the linear programming model:**\n- Soldering time constraint: Each Circuit Board requires 3 hours, each Microcontroller 4 hours, total soldering time available is 120 hours. So, $$3x + 4y \leq 120.$$\n- Quality inspection time constraint: Each Circuit Board requires 2 hours, each Microcontroller 1 hour, total inspection time available is 50 hours. So, $$2x + y \leq 50.$$\n- Warehouse space constraint: Total units produced cannot exceed 35, so $$x + y \leq 35.$$\n- Non-negativity constraints: $$x \geq 0, y \geq 0.$$\n- Objective function (profit to maximize): $$P = 20x + 30y.$$\n\n3. **Determine the optimal production quantities:**\nWe solve the system by checking the corner points of the feasible region defined by the inequalities.\n\n- Intersection of $$3x + 4y = 120$$ and $$2x + y = 50$$:\nMultiply second equation by 4: $$8x + 4y = 200.$$\nSubtract first equation: $$(8x + 4y) - (3x + 4y) = 200 - 120 \Rightarrow 5x = 80 \Rightarrow x = 16.$$\nSubstitute back: $$2(16) + y = 50 \Rightarrow 32 + y = 50 \Rightarrow y = 18.$$\n\n- Check if $$x + y \leq 35$$ at $(16,18)$: $$16 + 18 = 34 \leq 35,$$ so this point is feasible.\n\n- Other corner points to check:\n(a) $$x=0$$:\nFrom $$2(0) + y \leq 50 \Rightarrow y \leq 50,$$ and $$3(0) + 4y \leq 120 \Rightarrow y \leq 30,$$ and $$0 + y \leq 35 \Rightarrow y \leq 35.$$\nSo max $$y=30$$ at $(0,30)$.\n\n(b) $$y=0$$:\nFrom $$3x + 4(0) \leq 120 \Rightarrow x \leq 40,$$ $$2x + 0 \leq 50 \Rightarrow x \leq 25,$$ and $$x + 0 \leq 35 \Rightarrow x \leq 35.$$\nSo max $$x=25$$ at $(25,0)$.\n\n(c) Intersection of $$x + y = 35$$ and $$2x + y = 50$$:\nFrom $$x + y = 35 \Rightarrow y = 35 - x,$$ substitute into $$2x + y = 50$$:\n$$2x + 35 - x = 50 \Rightarrow x + 35 = 50 \Rightarrow x = 15,$$ then $$y = 20.$$\nCheck soldering constraint: $$3(15) + 4(20) = 45 + 80 = 125 > 120,$$ so not feasible.\n\n(d) Intersection of $$x + y = 35$$ and $$3x + 4y = 120$$:\nFrom $$x + y = 35 \Rightarrow y = 35 - x,$$ substitute into $$3x + 4y = 120$$:\n$$3x + 4(35 - x) = 120 \Rightarrow 3x + 140 - 4x = 120 \Rightarrow -x + 140 = 120 \Rightarrow x = 20,$$ then $$y = 15.$$\nCheck inspection constraint: $$2(20) + 15 = 40 + 15 = 55 > 50,$$ so not feasible.\n\n4. **Evaluate profit at feasible corner points:**\n- At $(0,30)$: $$P = 20(0) + 30(30) = 900.$$\n- At $(25,0)$: $$P = 20(25) + 30(0) = 500.$$\n- At $(16,18)$: $$P = 20(16) + 30(18) = 320 + 540 = 860.$$\n\n5. **Conclusion:**\nThe maximum profit is $$900$$ at producing $$0$$ Circuit Boards and $$30$$ Microcontrollers.\n\n**Final answers:**\n- 3.1 Linear programming model:\n$$\text{Maximize } P = 20x + 30y$$\n$$\text{subject to } 3x + 4y \leq 120, 2x + y \leq 50, x + y \leq 35, x \geq 0, y \geq 0.$$\n- 3.2 Optimal production: $$x=0$$ Circuit Boards, $$y=30$$ Microcontrollers.\n- 3.3 Maximum profit: $$900.$$