Hey guys stumped on this one \HELP PLEAZE A.S.A.P.
minimize p= 15x +18y
subject to x+2y<=20
3x + 2y>= 36
x>=0
y>=0
To solve this problem, we will use the method of linear programming. Linear programming involves optimizing (maximizing or minimizing) a linear objective function subject to linear constraints.
In this case, you are asked to minimize the objective function p = 15x + 18y, given the following constraints:
1. x + 2y ≤ 20
2. 3x + 2y ≥ 36
3. x ≥ 0
4. y ≥ 0
To solve this, you can follow these steps:
Step 1: Graph the Constraint Lines
Plot the boundary lines of each constraint on a coordinate plane. To do this, convert each inequality to an equation and plot the corresponding lines. Shade the region that satisfies the constraint.
Constraint 1: x + 2y ≤ 20
Rearrange the equation to get: 2y ≤ -x + 20
Plot the line: y = (-x + 20) / 2
Shade the region below the line.
Constraint 2: 3x + 2y ≥ 36
Rearrange the equation to get: 2y ≥ -3x + 36
Plot the line: y = (-3x + 36) / 2
Shade the region above the line.
Step 2: Identify the Feasible Region
The feasible region is the intersection of the shaded regions from the two constraints. It represents the region that satisfies all the given inequalities. Identify the overlapping region on the graph.
Step 3: Determine the Corner Points
The corner points of the feasible region are the vertices of the overlapping area. Find the coordinates of all the corner points.
Step 4: Evaluate the Objective Function
Plug in the coordinates of each corner point into the objective function p = 15x + 18y, and calculate the value of p for each point.
Step 5: Find the Minimum Value of the Objective Function
Compare the values of p for all corner points and determine the minimum value. This will give you the minimum value of p and the corresponding values of x and y that achieve it.
I hope this explanation helps you approach and solve this linear programming problem. Let me know if you have any further questions!