Applications of Derivatives
Linear Programming Problem
Grade 12
Question:
<p>The corner points of the feasible region determined by the system of linear constraints are \((0, 0)\), \((0, 40)\), \((20, 40)\), \((60, 20)\), \((60, 0)\). The objective function is \(z = 4x + 3y\).</p><p>Compare the quantity in Column A and Column B:</p><table border='1'><tr><th>Column A</th><th>Column B</th></tr><tr><td>Maximum of \(z\)</td><td>325</td></tr></table>
<p>The quantity in column A is greater</p>
<p>The quantity in column B is greater</p>
<p>The two quantities are equal</p>
<p>The relationship cannot be determined on the basis of the information Supplied</p>
Step-by-Step Solution
Key Concept: In linear programming, the maximum (or minimum) of the objective function always occurs at a corner point of the feasible region. Evaluate z = 4x + 3y at each corner point and identify the largest value.
<p><strong>Step 1:</strong> Evaluate the objective function z = 4x + 3y at each corner point.</p><p><strong>Step 2:</strong> At (0, 0): z = 4(0) + 3(0) = 0</p><p>At (0, 40): z = 4(0) + 3(40) = 120</p><p>At (20, 40): z = 4(20) + 3(40) = 80 + 120 = 200</p><p>At (60, 20): z = 4(60) + 3(20) = 240 + 60 = 300</p><p>At (60, 0): z = 4(60) + 3(0) = 240</p><p><strong>Step 3:</strong> Maximum value of z = 300 at point (60, 20)</p><p><strong>Step 4:</strong> Compare Column A (300) with Column B (325): 300 < 325</p><p>∴ Answer: A (Column B is greater)</p>
Correct Answer: A