<p>The feasible region is bounded. The maximum value of <em>z</em> = <em>4x</em> + <em>3y</em> is to be found. The corner points are (0, 0), (25, 0), (16, 16), and (0, 24). Find the maximum value of <em>z</em>.</p>
Step-by-Step Solution
Key Concept: In linear programming, the maximum (or minimum) of a linear objective function over a bounded feasible region always occurs at a corner point of the feasible region. Evaluate the objective function at each corner point and select the largest value.
<p><strong>Step 1:</strong> Identify all corner points of the feasible region: (0, 0), (25, 0), (16, 16), and (0, 24).</p><p><strong>Step 2:</strong> Evaluate the objective function z = 4x + 3y at each corner point.</p><ul><li>At (0, 0): z = 4(0) + 3(0) = 0</li><li>At (25, 0): z = 4(25) + 3(0) = 100</li><li>At (16, 16): z = 4(16) + 3(16) = 64 + 48 = 112</li><li>At (0, 24): z = 4(0) + 3(24) = 72</li></ul><p><strong>Step 3:</strong> Compare all values: 0, 100, 112, and 72.</p><p><strong>Step 4:</strong> The maximum value is 112, which occurs at the corner point (16, 16).</p><p>∴ Maximum value of z = <strong>112</strong></p>
Correct Answer: 112