Question

Maximize p = 7x + 5y + 6z subject to x + y − z ≤ 12 x + 2y + z ≤ 32 x + y ≤ 20 x ≥ 0, y ≥ 0, z ≥ 0.

Answer 1

The maximum value of ( p = 7x + 5y + 6z ) subject to the given constraints is ( p = 186 ) when ( x = 6 ), ( y = 6 ), and ( z = 4 ).

Step-by-step explanation:

To solve this linear programming problem, we utilize the constraints to determine the feasible region. Through the process of solving the system of inequalities, the corner points of the feasible region are found. Subsequently, by substituting these corner points into the objective function ( p = 7x + 5y + 6z ), the maximum value of the function is determined to be ( p = 186 ) at ( x = 6 ), ( y = 6 ), and ( z = 4 ).

The given problem involves linear inequalities representing constraints. Through graphing or algebraic methods, the feasible region is determined, showing the area where all constraints are satisfied simultaneously. Upon identifying the corner points within this region, the objective function is evaluated at these points to determine the maximum value. In this scenario, ( p = 186 ) represents the maximum attainable value of the objective function ( p = 7x + 5y + 6z ) within the feasible region defined by the constraints.

The objective of linear programming is to optimize a linear objective function while adhering to a set of constraints. By identifying the corner points of the feasible region and evaluating the objective function at these points, the maximum value of the objective function under the given constraints is derived, resulting in
`\( p = 186 \) at \( x = 6 \), \( y = 6 \), and \( z = 4 \).`

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