To solve this linear programming problem, you need to find the minimum and maximum values of the objective function P=-30x+25y, subject to the given constraints.
First, let's graph the constraints on a coordinate plane:
1. Constraint: 2x+3y>=30
- When x=0, y=10
- When x=15, y=0
The area above this line is in our region.
2. Constraint: 2x+y<=26
- When x=0, y=26
- When x=13, y=0
The area below this line is in our region.
3. Constraint: -6x+5y<=50
- When x=0, y=10
- When x=-50/6 or -8 1/3, y=0
The area below this line is in our region.
Based on these constraints, we can see that our region of interest is a triangle from (0,10) to the intersection of constraint #2 and constraint #3 down to the intersection of constraint #1 and constraint #2.
To find the intersection points, solve the equations for the corresponding constraint pairs:
1. Constraint #1 and constraint #2:
2x+3y=30
2x+y=26
Solving this system of equations, we get:
2y=4 or y=2
From the second equation, x=12
Therefore, the intersection point is (12,2).
2. Constraint #2 and constraint #3:
2x+y=26
-6x+5y=50
Multiplying the first equation by 3 to eliminate y, we get:
6x+3y=78
Subtracting this equation from the second equation, we get:
-8y=-128
Solving for y, we get y=16
Substituting y=16 into the first equation, we get x=5
Therefore, the intersection point is (5,16).
Now, calculate the value of P at each of the intersection points: (5,16), (12,2), and (0,10). Plug the x and y values into the objective function P=-30x+25y, and compute the corresponding P values.
Choose the either the minimum or maximum P value, depending on what you are looking for (e.g., profit).