1. **Restate the problem:** We need to determine how the 100 acres should be used for planting crops A, B, and C by applying the vertex method to the feasible region defined by the constraints. 2. **Recall the vertex method:** The vertex method involves evaluating the objective function at each vertex (corner point) of the feasible region. The maximum or minimum value of the objective function occurs at one of these vertices. 3. **Identify the objective function:** From earlier parts, the profit function is: $$\text{Profit} = 60x + 75y + 25z$$ where $x$, $y$, and $z$ represent acres of crops A, B, and C respectively. 4. **Use the land constraint:** Since all 100 acres are used, $$x + y + z = 100 \implies z = 100 - x - y$$ 5. **Rewrite the profit function in terms of $x$ and $y$ only:** $$\text{Profit} = 60x + 75y + 25(100 - x - y) = 60x + 75y + 2500 - 25x - 25y = 35x + 50y + 2500$$ 6. **Evaluate profit at each vertex of the feasible region $R$:** Vertices found in part (d)(i) are (assumed known): - $V_1 = (25, 33.75)$ - $V_2 = (36, 64)$ - $V_3 = (42, 58)$ - $V_4 = (25, 0)$ Calculate profit at each: - At $V_1$: $35(25) + 50(33.75) + 2500 = 875 + 1687.5 + 2500 = 5062.5$ - At $V_2$: $35(36) + 50(64) + 2500 = 1260 + 3200 + 2500 = 6960$ - At $V_3$: $35(42) + 50(58) + 2500 = 1470 + 2900 + 2500 = 6870$ - At $V_4$: $35(25) + 50(0) + 2500 = 875 + 0 + 2500 = 3375$ 7. **Determine the maximum profit:** The maximum profit is $6960$ at vertex $V_2 = (36, 64)$. 8. **Interpretation:** The farmer should plant 36 acres of crop A, 64 acres of crop B, and $100 - 36 - 64 = 0$ acres of crop C to maximize profit. **Final answer:** - Plant 36 acres of crop A - Plant 64 acres of crop B - Plant 0 acres of crop C - Maximum expected profit is 6960