1. **Problem Statement:** We want to determine how many REGULAR and SUPER tents 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: $1x + 2y \leq 32$ - Assembly department labor hours: $3x + 4y \leq 84$ - Demand limit for SUPER tents: $y \leq 12$ - Non-negativity: $x \geq 0, y \geq 0$ 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 Intersection Points (Vertices):** - Intersection of $x + 2y = 32$ and $3x + 4y = 84$: Multiply first by 3: $3x + 6y = 96$ Subtract second: $(3x + 6y) - (3x + 4y) = 96 - 84 \Rightarrow 2y = 12 \Rightarrow y = 6$ Substitute $y=6$ into $x + 2(6) = 32 \Rightarrow x + 12 = 32 \Rightarrow x = 20$ Vertex: $(20,6)$ - Intersection of $x + 2y = 32$ and $y = 12$: Substitute $y=12$: $x + 2(12) = 32 \Rightarrow x + 24 = 32 \Rightarrow x = 8$ Vertex: $(8,12)$ - Intersection of $3x + 4y = 84$ and $y = 12$: Substitute $y=12$: $3x + 4(12) = 84 \Rightarrow 3x + 48 = 84 \Rightarrow 3x = 36 \Rightarrow x = 12$ Vertex: $(12,12)$ - Intersections with axes: - $x$-axis: $y=0$ - From $x + 2y \leq 32$: $x \leq 32$ - From $3x + 4y \leq 84$: $3x \leq 84 \Rightarrow x \leq 28$ So max $x$ on $x$-axis is $28$. - $y$-axis: $x=0$ - From $x + 2y \leq 32$: $2y \leq 32 \Rightarrow y \leq 16$ - From $3x + 4y \leq 84$: $4y \leq 84 \Rightarrow y \leq 21$ - Demand limit: $y \leq 12$ So max $y$ on $y$-axis is $12$. 7. **Evaluate Profit at Vertices:** - At $(0,0)$: $P=50(0)+80(0)=0$ - At $(28,0)$: $P=50(28)+80(0)=1400$ - At $(0,12)$: $P=50(0)+80(12)=960$ - At $(20,6)$: $P=50(20)+80(6)=1000+480=1480$ - At $(8,12)$: $P=50(8)+80(12)=400+960=1360$ - At $(12,12)$: $P=50(12)+80(12)=600+960=1560$ 8. **Optimal Solution:** Maximum profit is $1560$ at $(12,12)$. **Answer:** - Manufacture 12 REGULAR tents and 12 SUPER tents weekly. - Maximum profit is $1560$.