Step 1: Understanding the Concept:
Linear Programming Problems (LPP) seek to find the best outcome (such as maximum profit or minimum cost) in a given mathematical model whose requirements are represented by linear relationships. The "Corner Point Theorem" states that the maximum or minimum value of an objective function always occurs at one of the vertices (corner points) of the feasible region formed by the constraints.
Key Formula or Approach:
1. Identify the feasible region using the inequalities.
2. Find the coordinates of the vertices.
3. Substitute these coordinates into the objective function \(Z = 3x + 4y\).
Step 2: Detailed Explanation:
The constraints \(x \geq 0\) and \(y \geq 0\) limit our search to the first quadrant.
The constraint \(x + y \leq 4\) is a line crossing the axes. Let's find the intercepts for \(x + y = 4\):
Setting \(y = 0\) gives \(x = 4\). Point: \((4, 0)\).
Setting \(x = 0\) gives \(y = 4\). Point: \((0, 4)\).
The region is a triangle bounded by \((0, 0)\), \((4, 0)\), and \((0, 4)\).
Evaluate \(Z = 3x + 4y\) at these corners:
- At \((0, 0)\): \(Z = 3(0) + 4(0) = 0\)
- At \((4, 0)\): \(Z = 3(4) + 4(0) = 12\)
- At \((0, 4)\): \(Z = 3(0) + 4(4) = 16\)
The maximum value is \(16\).
Step 3: Final Answer:
Testing the vertices of the feasible region, we find the maximum value of the objective function is \(16\) at point \((0, 4)\).