1. **State the problem:** We want to determine how many shares of Company 1 (x) and Company 2 (y) the investor should buy to maximize return given investment constraints. 2. **Define variables:** Let $x$ = number of shares bought of Company 1. Let $y$ = number of shares bought of Company 2. 3. **Objective function:** Price increase for Company 1 is from 40 to 55, so gain per share = $55 - 40 = 15$. Price increase for Company 2 is from 25 to 43, so gain per share = $43 - 25 = 18$. Maximize total return: $$ R = 15x + 18y $$ 4. **Constraints translate from problem statement:** - Investment limit: $40x + 25y \leq 50000$ - Minimum investment in Company 1: $40x \geq 15000 \implies x \geq 375$ - Minimum investment in Company 2: $25y \geq 10000 \implies y \geq 400$ - Maximum investment in Company 2 due to risk: $25y \leq 25000 \implies y \leq 1000$ - Also, $x, y \geq 0$ (number of shares cannot be negative) 5. **Summarize the linear programming model:** Maximize: $$ R = 15x + 18y $$ Subject to: $$40x + 25y \leq 50000$$ $$x \geq 375$$ $$400 \leq y \leq 1000$$ $$x \geq 0, y \geq 0$$ 6. **Graph the feasible region:** - Plot the lines corresponding to each inequality. - The feasible region is the polygon bounded by those constraints, limited by the investment total, and the minimum/maximum share constraints on $x$ and $y$. 7. **Find coordinates of each extreme point (vertex) of the feasible region:** Calculate intersections of: - $x=375$ and $y=400$ - $x=375$ and $40x + 25y = 50000$ - $y=1000$ and $40x + 25y = 50000$ - $y=1000$ and $x$ at minimum or non-negativity bound Check all combinations within constraints. Calculate: - Intersection of $x=375$ and $40(375) + 25y = 50000$: $$40 \times 375 + 25y = 50000 \Rightarrow 15000 + 25y = 50000 \Rightarrow 25y = 35000 \Rightarrow y = 1400$$ But $y$ can't exceed 1000, so this point is invalid due to the $y \leq 1000$ constraint. - Intersection of $y=1000$ and $40x + 25 \times 1000 = 50000$: $$40x + 25000 = 50000 \Rightarrow 40x = 25000 \Rightarrow x = 625$$ This satisfies $x \geq 375$, valid point. - Intersection of $x=375$ and $y=400$: Valid and satisfies all constraints. - Intersection of $y=400$ and $40x + 25 \times 400 = 50000$: $$40x + 10000 = 50000 \Rightarrow 40x = 40000 \Rightarrow x = 1000$$ This satisfies $x \geq 375$, valid point. 8. **List vertices:** - $A = (375, 400)$ - $B = (375, 1000)$ (Check if feasible: we must ensure $40\times 375 + 25\times 1000 = 15000 + 25000 = 40000 \leq 50000$ valid) - $C = (625, 1000)$ - $D = (1000, 400)$ 9. **Find the optimal solution by evaluating return $R = 15x + 18y$ at each vertex:** - At $A$: $R = 15 \times 375 + 18 \times 400 = 5625 + 7200 = 12825$ - At $B$: $R = 15 \times 375 + 18 \times 1000 = 5625 + 18000 = 23625$ - At $C$: $R = 15 \times 625 + 18 \times 1000 = 9375 + 18000 = 27375$ - At $D$: $R = 15 \times 1000 + 18 \times 400 = 15000 + 7200 = 22200$ Maximum return is $27375$ at point $C = (625, 1000)$. 10. **Interpretation:** Buy 625 shares of Company 1 and 1000 shares of Company 2 to maximize profit within constraints. ---