Sets, Relations & Functions
Linear Programming
Grade 11
Question:
<p>The maximum value of <em>z</em> will occur at a corner point of the feasible region, where <em>z</em> = <em>3x</em> − <em>4y</em>. The corner points are (0, 0), (12, 6), and (0, 4). Find the maximum value of <em>z</em>.</p>
Step-by-Step Solution
Key Concept: Linear programming extrema occur at vertices of the convex feasible region. Evaluate the objective function at each corner point and compare to find the maximum.
<p><strong>Step 1:</strong> Identify the corner points of the feasible region: (0, 0), (12, 6), and (0, 4).</p><p><strong>Step 2:</strong> Evaluate z = 3x − 4y at each corner point:</p><ul><li>At (0, 0): z = 3(0) − 4(0) = 0</li><li>At (12, 6): z = 3(12) − 4(6) = 36 − 24 = 12</li><li>At (0, 4): z = 3(0) − 4(4) = −16</li></ul><p><strong>Step 3:</strong> Compare all values: 0, 12, and −16.</p><p>∴ The maximum value of z is <strong>12</strong>, occurring at the corner point (12, 6).</p>
Correct Answer: 12