1. **Problem Statement:** We want to determine the number of shares of Company 1 and Company 2 an investor should buy to maximize returns while satisfying investment constraints. 2. **Define Variables:** Let $x$ = number of shares of Company 1. Let $y$ = number of shares of Company 2. 3. **Prices and Constraints:** Price per share Company 1 = $40$ Price per share Company 2 = $25$ Minimum investment in Company 1 is $15,000$, so $$40x \geq 15000 \implies x \geq \frac{15000}{40} = 375.$$ Minimum investment in Company 2 is $10,000$, so $$25y \geq 10000 \implies y \geq \frac{10000}{25} = 400.$$ Maximum investment in Company 2 is $25,000$, so $$25y \leq 25000 \implies y \leq 1000.$$ Maximum total investment is $50,000$, so $$40x + 25y \leq 50000.$$ 4. **Objective Function:** Returns per share for Company 1 are estimated to increase from $40$ to $55$, so profit per share is $55 - 40 = 15$. Returns per share for Company 2 are estimated to increase from $25$ to $43$, so profit per share is $43 - 25 = 18$. Maximize total return: $$Z = 15x + 18y.$$ 5. **Summary of linear programming model:** \[\begin{cases} \text{Maximize } Z= 15x + 18y \\ \text{subject to: } \\ x \geq 375 \\ y \geq 400 \\ y \leq 1000 \\ 40x + 25y \leq 50000 \\ x, y \geq 0 \end{cases}\] 6. **Graph the feasible region:** The feasible region is bounded by the inequalities above. It lies in the first quadrant with $x \geq 375$, $400 \leq y \leq 1000$, and below the line $$40x + 25y = 50000.$$ 7. **Find extreme points (corner points) by considering intersections:** - $A = (375, 400)$ (minimum investments in both) - $B = (375, 1000)$ (min invest in 1 and max in 2) - $C$: Intersection of $y=1000$ and $40x + 25y = 50000$ $$40x + 25(1000) = 50000 \implies 40x + 25000 = 50000 \implies 40x = 25000 \implies x=625.$$ So $C=(625,1000)$ - $D$: Intersection of $x=375$ and $40x + 25y = 50000$ $$40(375) + 25y = 50000 \implies 15000 + 25y = 50000 \implies 25y=35000 \implies y=1400,$ but $y=1400$ violates $y \leq 1000$, so discard - $E$: Intersection of $y=400$ and $40x + 25y = 50000$ $$40x + 25(400) = 50000 \implies 40x + 10000 = 50000 \implies 40x = 40000 \implies x=1000.$$ So $E=(1000,400)$ 8. **Evaluate objective $Z=15x+18y$ at extreme points:** - $A: Z = 15(375) + 18(400) = 5625 + 7200 = 12825$ - $B: Z = 15(375) + 18(1000) = 5625 + 18000 = 23625$ - $C: Z = 15(625) + 18(1000) = 9375 + 18000 = 27375$ - $E: Z = 15(1000) + 18(400) = 15000 + 7200 = 22200$ 9. **Optimal Solution:** The maximum return is at point $C = (625, 1000)$ with total return $Z = 27375$. This means investing in 625 shares of Company 1 and 1000 shares of Company 2 maximizes returns within constraints.