Find the maximum value of the objective function z=24x+7y subject to the following constraints : 0¡Üx¡Ü10, 0¡Üy¡Ü5,3x+2y¡Ý6.
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Given the objective function C=3x−2y and constraints x≥0, y≥0, 2x+y≤10, 3x+2y≤18, identify the corner point at which the maximum value of C occurs
To find the maximum value of the objective function z=24x+7y subject to the given constraints, we can use the method of linear programming.
1. Graph the feasible region:
Plot the inequalities 0≤x≤10, 0≤y≤5, and 3x+2y≥6 on a coordinate plane. The feasible region is the shaded area where all constraints are satisfied.
2. Find the vertices of the feasible region:
The feasible region is bounded by the lines x=0, x=10, y=0, y=5, and 3x+2y=6. Find the points of intersection of these lines to determine the vertices of the feasible region.
By solving the equations, we find the following vertices:
A: (0, 3)
B: (0, 5)
C: (2, 4)
D: (4, 3)
E: (6, 0)
3. Evaluate the objective function at each vertex:
Substitute the x and y values of each vertex into the objective function z=24x+7y to find the corresponding values of z for each vertex.
A: z = 24(0) + 7(3) = 21
B: z = 24(0) + 7(5) = 35
C: z = 24(2) + 7(4) = 80
D: z = 24(4) + 7(3) = 105
E: z = 24(6) + 7(0) = 144
4. Compare the values of z for each vertex:
Since the objective is to maximize z, we compare the values of z obtained at each vertex.
The maximum value of z is 144, which occurs at the point (6, 0) or vertex E.
Therefore, the maximum value of the objective function z=24x+7y subject to the given constraints is 144, and it occurs when x=6 and y=0.