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Graph the system of constraints for the following linear programming problem. Name all vertices. Find the values of x and y that maximize the objective function C = -4x + 5y. Then find the maximum value.

Constraints:
y ≤ -1/2x + 5
x ≥ 0
y ≥ 0
X = ?
Y = ?
Maximum = ?

User TKumar
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1 Answer

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Final answer:

To solve the given linear programming problem, plot the constraint lines, shade the feasible region, identify vertices, and evaluate the objective function at each vertex to find the maximum value of C. The optimal values for x and y, along with the maximum value of C, will be determined in this way.

Step-by-step explanation:

To graph the system of constraints for the linear programming problem with the objective function C = -4x + 5y, and constraints y ≤ -1/2x + 5, x ≥ 0, and y ≥ 0, start by plotting the constraint lines on a graph. The line y = -1/2x + 5 should be drawn with a slope of -1/2 and a y-intercept at (0,5). Because y is less than or equal to this value, shade below the line. x ≥ 0 means we are only considering the positive x-axis, and similarly, y ≥ 0 means we only consider the positive y-axis. The area of the graph that is shaded by all constraints represents the feasible region.

To find the vertices of the feasible region, identify the points of intersection of the constraint lines and the axes. Label these points on the graph. With the constraints given, the vertices should be at (0,0), (10,0) representing the x-intercept of the constraint line, and (0,5) representing the y-intercept of the constraint line.

Next, evaluate the objective function C at each vertex to find which gives the maximum and minimum values. Since we want to maximize C, we substitute the x and y values from the vertices into C and determine which vertex results in the highest value of C. This vertex provides the optimal solution for x and y, and the corresponding value of C is the maximum value.

User Aapo Kyrola
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