1. **Problem Statement:** MetroLink Seating is designing a seating layout for a passenger car with Premium and Standard sections. Each Premium section yields a profit of 600 and requires 4 meters. Each Standard section yields a profit of 250 and requires 2 meters. The total cabin length is 20 meters. Constraints include: for every Premium section, at least 2 Standard sections must be included, at least 3 Standard sections must be included overall, and at most 4 Premium sections can be included. 2. **Define Variables:** Let $x$ = number of Premium sections. Let $y$ = number of Standard sections. 3. **Constraints:** - Length constraint: $4x + 2y \leq 20$ - Market policy 1: $y \geq 2x$ - Market policy 2: $y \geq 3$ - Market policy 3: $x \leq 4$ - Non-negativity: $x \geq 0$, $y \geq 0$ 4. **Objective Function:** Maximize profit $P = 600x + 250y$ 5. **Graphing Constraints:** - Line 1: $4x + 2y = 20$ or $y = 10 - 2x$ - Line 2: $y = 2x$ - Line 3: $y = 3$ - Line 4: $x = 4$ The feasible region is the intersection of all inequalities: $$\begin{cases} 4x + 2y \leq 20 \\ y \geq 2x \\ y \geq 3 \\ x \leq 4 \\ x,y \geq 0 \end{cases}$$ 6. **Find Corner Points of Feasible Region:** - Intersection of $y=2x$ and $y=3$: set $2x=3 \Rightarrow x=1.5$, $y=3$ - Intersection of $y=2x$ and $4x+2y=20$: substitute $y=2x$ into $4x+2(2x)=20 \Rightarrow 4x+4x=20 \Rightarrow 8x=20 \Rightarrow x=2.5$, $y=5$ - Intersection of $y=3$ and $4x+2y=20$: substitute $y=3$ into $4x+2(3)=20 \Rightarrow 4x+6=20 \Rightarrow 4x=14 \Rightarrow x=3.5$, $y=3$ - Intersection of $x=4$ and $y=2x$: $x=4$, $y=8$ (check if feasible: $4(4)+2(8)=16+16=32 > 20$ not feasible) - Intersection of $x=4$ and $y=3$: $x=4$, $y=3$ (check feasibility: $4(4)+2(3)=16+6=22 > 20$ not feasible) 7. **Evaluate Objective at Feasible Corner Points:** - At $(1.5,3)$: $P=600(1.5)+250(3)=900+750=1650$ - At $(2.5,5)$: $P=600(2.5)+250(5)=1500+1250=2750$ - At $(3.5,3)$: $P=600(3.5)+250(3)=2100+750=2850$ 8. **Optimal Solution:** The maximum profit is $2850$ at $x=3.5$ Premium sections and $y=3$ Standard sections. Since sections must be whole numbers, check integer points near $(3.5,3)$: - $(3,4)$: check constraints and profit - Length: $4(3)+2(4)=12+8=20 \leq 20$ OK - $y \geq 2x$: $4 \geq 6$ No, fails - $(4,3)$: - Length: $4(4)+2(3)=16+6=22 > 20$ No - $(3,3)$: - Length: $4(3)+2(3)=12+6=18 \leq 20$ OK - $y \geq 2x$: $3 \geq 6$ No - $(2,4)$: - Length: $4(2)+2(4)=8+8=16 \leq 20$ OK - $y \geq 2x$: $4 \geq 4$ OK - $y \geq 3$: OK - $x \leq 4$: OK - Profit: $600(2)+250(4)=1200+1000=2200$ Thus, the best integer solution satisfying all constraints is $x=2$, $y=4$ with profit $2200$.