1. **Problem Statement:** We want to determine how many REGULAR tents ($x$) and SUPER tents ($y$) to manufacture weekly to maximize profit, given labor hour constraints and demand limits. 2. **Define Variables:** Let $x$ = number of REGULAR tents Let $y$ = number of SUPER tents 3. **Constraints:** - Cutting department labor hours: each REGULAR tent requires 1 hour, each SUPER tent 2 hours, max 32 hours $$x + 2y \leq 32$$ - Assembly department labor hours: each REGULAR tent requires 3 hours, each SUPER tent 4 hours, max 84 hours $$3x + 4y \leq 84$$ - Demand limit for SUPER tents: $$y \leq 12$$ - Non-negativity: $$x \geq 0, \quad y \geq 0$$ 4. **Objective Function (Profit):** Profit per REGULAR tent = $160 - 110 = 50$ Profit per SUPER tent = $210 - 130 = 80$ Maximize: $$P = 50x + 80y$$ 5. **Graphical Method:** Plot the constraints and find the feasible region. 6. **Find corner points of feasible region:** - Intersection of $x + 2y = 32$ and $3x + 4y = 84$: Multiply first by 2: $$2x + 4y = 64$$ Subtract from second: $$3x + 4y - (2x + 4y) = 84 - 64 \Rightarrow x = 20$$ Substitute $x=20$ into $x + 2y = 32$: $$20 + 2y = 32 \Rightarrow 2y = 12 \Rightarrow y = 6$$ Point: $(20,6)$ - Intersection of $x + 2y = 32$ and $y = 12$: $$x + 2(12) = 32 \Rightarrow x + 24 = 32 \Rightarrow x = 8$$ Point: $(8,12)$ - Intersection of $3x + 4y = 84$ and $y = 12$: $$3x + 4(12) = 84 \Rightarrow 3x + 48 = 84 \Rightarrow 3x = 36 \Rightarrow x = 12$$ Point: $(12,12)$ - Intercepts: - $x$-intercept of $x + 2y = 32$ is $(32,0)$ - $x$-intercept of $3x + 4y = 84$ is $(28,0)$ - $y$-intercept of $x + 2y = 32$ is $(0,16)$ but $y \leq 12$ limits it to $(0,12)$ - $y$-intercept of $3x + 4y = 84$ is $(0,21)$ but $y \leq 12$ limits it to $(0,12)$ 7. **Evaluate profit at feasible corner points:** - At $(0,0)$: $P=0$ - At $(32,0)$ (check feasibility): $3(32)+4(0)=96 > 84$ not feasible - At $(28,0)$ (feasible for assembly but check cutting): $28 + 2(0) = 28 \leq 32$ feasible Profit: $50(28)+80(0)=1400$ - At $(20,6)$: Profit: $50(20)+80(6)=1000+480=1480$ - At $(8,12)$: Profit: $50(8)+80(12)=400+960=1360$ - At $(12,12)$ (check cutting): $12 + 2(12) = 12 + 24 = 36 > 32$ not feasible 8. **Conclusion:** Maximum profit is $1480$ at $(x,y) = (20,6)$. **Final answer:** The company should manufacture 20 REGULAR tents and 6 SUPER tents weekly to maximize profit of 1480.