1. **Problem Statement:** An investor wants to invest in two companies: Company 1 (extractive) and Company 2 (tech). Prices per share are $40$ for Company 1 and $25$ for Company 2. Expected future prices are $55$ and $43$ respectively. The investor has a budget of $50,000$, wants to invest at least $15,000$ in Company 1 and at least $10,000$ in Company 2, and at most $25,000$ in Company 2 due to higher risk. The goal is to maximize total return. 2. **Define variables:** Let $x$ = number of shares bought of Company 1. Let $y$ = number of shares bought of Company 2. 3. **Constraints:** - Budget constraint: $40x + 25y \leq 50,000$ - Minimum investment in Company 1: $40x \geq 15,000 \Rightarrow x \geq \frac{15,000}{40} = 375$ - Minimum investment in Company 2: $25y \geq 10,000 \Rightarrow y \geq \frac{10,000}{25} = 400$ - Maximum investment in Company 2: $25y \leq 25,000 \Rightarrow y \leq 1,000$ - Non-negativity: $x \geq 0$, $y \geq 0$ 4. **Objective function (maximize total return):** Return from Company 1 per share = $55 - 40 = 15$ Return from Company 2 per share = $43 - 25 = 18$ Maximize $Z = 15x + 18y$ --- 5. **Summary of the linear programming model:** Maximize $$Z = 15x + 18y$$ subject to: $$40x + 25y \leq 50,000$$ $$x \geq 375$$ $$y \geq 400$$ $$y \leq 1,000$$ $$x, y \geq 0$$ --- 6. **Graphing the feasible region:** The feasible region is bounded by the lines: - $40x + 25y = 50,000$ - $x = 375$ - $y = 400$ - $y = 1,000$ 7. **Find extreme points (corners) by solving intersections:** - Intersection of $x=375$ and $y=400$: $(375, 400)$ - Intersection of $x=375$ and $y=1,000$: $(375, 1,000)$ - Intersection of $y=400$ and $40x + 25y = 50,000$: $$40x + 25(400) = 50,000 \Rightarrow 40x + 10,000 = 50,000 \Rightarrow 40x = 40,000 \Rightarrow x = 1,000$$ So point is $(1,000, 400)$ - Intersection of $y=1,000$ and $40x + 25y = 50,000$: $$40x + 25(1,000) = 50,000 \Rightarrow 40x + 25,000 = 50,000 \Rightarrow 40x = 25,000 \Rightarrow x = 625$$ So point is $(625, 1,000)$ 8. **Evaluate objective function $Z$ at each extreme point:** - At $(375, 400)$: $$Z = 15(375) + 18(400) = 5,625 + 7,200 = 12,825$$ - At $(375, 1,000)$: $$Z = 15(375) + 18(1,000) = 5,625 + 18,000 = 23,625$$ - At $(1,000, 400)$: $$Z = 15(1,000) + 18(400) = 15,000 + 7,200 = 22,200$$ - At $(625, 1,000)$: $$Z = 15(625) + 18(1,000) = 9,375 + 18,000 = 27,375$$ 9. **Optimal solution:** The maximum return is $27,375$ at $(625, 1,000)$. **Interpretation:** - Buy $625$ shares of Company 1. - Buy $1,000$ shares of Company 2. - This respects all constraints and maximizes the total return.