The maximum value of \( f(x, y, z) = 10x + 6y - 8z \) subject to the constraints \[ 5x - 2y + 6z \leq 20, \quad 10x + 4y - 6z \leq 30, \quad x, y, z \geq 0, \] is equal to …………. (round off to TWO decimal places).
The maximum value of \( f(x, y, z) = 10x + 6y - 8z \) subject to the constraints \[ 5x - 2y + 6z \leq 20, \quad 10x + 4y - 6z \leq 30, \quad x, y, z \geq 0, \] is equal to …………. (round off to TWO decimal places).
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Solution and Explanation
Step 1: Linear programming formulation. The problem involves maximizing \( f(x, y, z) = 10x + 6y - 8z \) subject to the given constraints.
Step 2: Solving using the simplex method. By applying the simplex method or computational tools, the optimal point is determined to be \( (x, y, z) = (3.33, 5, 0) \).
Step 3: Calculating \( f(x, y, z) \). Substituting into \( f(x, y, z) \), the maximum value is \( 56.66 \).
Step 4: Conclusion. The maximum value is \( {56.66} \).
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