This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Define variables and formulate the linear programming problem. Let C be the number of canals for corn. Let W be the number of canals for wheat. The objective is to maximize profit P. Profit from corn is Rs. 20 per canal. Profit from wheat is Rs. 15 per canal. P = 20C + 15W Constraints: 1. Total land: The farmer possesses 100 canals of land. C + W 100 2. Working hours: Corn requires 3 hours/canal, wheat requires 2 hours/canal. Total working hours cannot exceed 240. 3C + 2W 240 3. Non-negativity: The number of canals cannot be negative. C 0, W 0 Step 2: Find the corner points of the feasible region by considering the boundary lines of the constraints. The boundary lines are: 1. C + W = 100 2. 3C + 2W = 240 3. C = 0 (W-axis) 4. W = 0 (C-axis) The corner points are: Intersection of C=0 and W=0: (0, 0) Intersection of C=0 and C+W=100: (0, 100) Intersection of W=0 and 3C+2W=240: 3C + 2(0) = 240 3C = 240 C = 80. So, (80, 0). Intersection of C+W=100 and 3C+2W=240: From C+W=100, we get W = 100 - C. Substitute into the second equation: 3C + 2(100 - C) = 240 3C + 200 - 2C = 240 C + 200 = 240 C = 40 Now find W: W = 100 - 40 = 60. So, (40, 60). Step 3: Evaluate the profit function P = 20C + 15W at each corner point. At (0, 0): P = 20(0) + 15(0) = 0 At (0, 100): P = 20(0) + 15(100) = 1500 At (80, 0): P = 20(80) + 15(0) = 1600 At (40, 60): P = 20(40) + 15(60) = 800 + 900 = 1700 Step 4: Identify the maximum profit. The maximum profit is Rs. 1700, which occurs when C=40 and W=60. The farmer should cultivate 40 canals of corn and 60 canals of wheat to maximize profit. The maximum profit is Rs. 1700. Corn: 40 canals, Wheat: 60 canals, Maximum Profit: Rs. 1700 Send me the next one 📸