1. **State the problem:** An investor wants to invest in two companies, Company 1 (extractive) and Company 2 (tech). 2. **Given data:** - Company 1 price per share: $40 - Company 2 price per share: $25 - Expected Company 1 price after investment: $55 - Expected Company 2 price after investment: $43 - Total investment maximum: $50,000 - Minimum investment in Company 1: $15,000 - Minimum investment in Company 2: $10,000 - Maximum investment in Company 2 due to risk: $25,000 3. **Variables:** Let $x$ = amount invested in Company 1 Let $y$ = amount invested in Company 2 4. **Constraints:** $$x + y \leq 50000$$ $$x \geq 15000$$ $$10,000 \leq y \leq 25000$$ 5. **Objective:** Calculate the total number of shares bought in each company: $$\text{Shares in Company 1} = \frac{x}{40}$$ $$\text{Shares in Company 2} = \frac{y}{25}$$ 6. **Expected value after price increase:** $$\text{Value from Company 1 after increase} = \frac{x}{40} \times 55 = \frac{55x}{40} = 1.375x$$ $$\text{Value from Company 2 after increase} = \frac{y}{25} \times 43 = \frac{43y}{25} = 1.72y$$ 7. **Total expected value after investment:** $$V = 1.375x + 1.72y$$ 8. **Optimization problem:** Maximize $$V = 1.375x + 1.72y$$ with constraints above. 9. **Check corner points:** - At $x=15000$, $y=10000$, $$V=1.375(15000)+1.72(10000)=20625+17200=37825$$ - At $x=15000$, $y=25000$, $$V=1.375(15000)+1.72(25000)=20625+43000=63625$$ - At $x=25000$, $y=25000$ (max $y$ and max $x$ to not exceed $50,000$), $$V=1.375(25000)+1.72(25000)=34375+43000=77375$$ - At $x=40000$, $y=10000$ (max $x$ given minimum $y$), $$V=1.375(40000)+1.72(10000)=55000+17200=72200$$ 10. **Conclusion:** The maximum expected value occurs at $x=25000$, $y=25000$. The investor should invest $25,000 in Company 1 and $25,000 in Company 2 to maximize expected returns under given constraints.