1. **Problem Statement:** A jewelry store has 18 ounces of gold and 20 ounces of platinum. Each necklace requires 3 ounces of gold and 2 ounces of platinum. Each bracelet requires 2 ounces of gold and 4 ounces of platinum. The demand for bracelets is at most 4. Profit per necklace is 300 and per bracelet is 400. We want to find the number of necklaces ($x$) and bracelets ($y$) to maximize profit. 2. **Define variables:** Let $x$ = number of necklaces Let $y$ = number of bracelets 3. **Constraints:** - Gold constraint: $$3x + 2y \leq 18$$ - Platinum constraint: $$2x + 4y \leq 20$$ - Demand constraint for bracelets: $$y \leq 4$$ - Non-negativity: $$x \geq 0, y \geq 0$$ 4. **Objective function:** Maximize profit $$P = 300x + 400y$$ 5. **Graphical method:** Plot the constraints on the $xy$-plane: - $3x + 2y = 18$ - $2x + 4y = 20$ - $y = 4$ The feasible region is the intersection of all inequalities. 6. **Find corner points of the feasible region:** - Intersection of $3x + 2y = 18$ and $2x + 4y = 20$: Multiply first by 2: $6x + 4y = 36$ Subtract second: $(6x + 4y) - (2x + 4y) = 36 - 20 \Rightarrow 4x = 16 \Rightarrow x = 4$ Substitute $x=4$ into $3x + 2y = 18$: $3(4) + 2y = 18 \Rightarrow 12 + 2y = 18 \Rightarrow 2y = 6 \Rightarrow y = 3$ Point: $(4,3)$ - Intersection of $3x + 2y = 18$ and $y=4$: Substitute $y=4$: $3x + 2(4) = 18 \Rightarrow 3x + 8 = 18 \Rightarrow 3x = 10 \Rightarrow x = \frac{10}{3} \approx 3.33$ Point: $(3.33,4)$ - Intersection of $2x + 4y = 20$ and $y=4$: Substitute $y=4$: $2x + 4(4) = 20 \Rightarrow 2x + 16 = 20 \Rightarrow 2x = 4 \Rightarrow x = 2$ Point: $(2,4)$ - Intercepts: For $3x + 2y = 18$, intercepts are $(6,0)$ and $(0,9)$ For $2x + 4y = 20$, intercepts are $(10,0)$ and $(0,5)$ 7. **Evaluate profit at corner points:** - At $(0,0)$: $P=0$ - At $(6,0)$: $P=300(6)+400(0)=1800$ - At $(0,4)$: $P=300(0)+400(4)=1600$ - At $(2,4)$: $P=300(2)+400(4)=600+1600=2200$ - At $(4,3)$: $P=300(4)+400(3)=1200+1200=2400$ - At $(3.33,4)$: $P=300(3.33)+400(4)=999+1600=2599$ 8. **Check feasibility:** Point $(3.33,4)$ satisfies all constraints, so it is feasible. 9. **Conclusion:** Maximum profit is approximately 2599 when making about 3.33 necklaces and 4 bracelets. Since the number of necklaces must be whole, the store can consider 3 or 4 necklaces and 4 bracelets and check which yields better profit within constraints.