Solve the linear programming problem by the method of corners.
Find the minimum and maximum of
P = 3x + 2y subject to
3x + 5y ≥ 20
3x + y ≤ 16
−2x + y ≤ 2
x ≥ 0, y ≥ 0
The minimum is P =
at (x, y) =
To solve the linear programming problem using the method of corners, we need to find the feasible region and then check the value of the objective function at each corner point to determine the minimum and maximum.
Step 1: Graph the inequalities
The inequalities can be graphed on a coordinate plane. Start by graphing the equations:
3x + 5y = 20 (equation 1)
3x + y = 16 (equation 2)
-2x + y = 2 (equation 3)
To graph each of these equations, we can find their respective intercepts by setting one variable to zero and solving for the other variable.
For equation 1:
Let x = 0, then 5y = 20, y = 4; hence, one point on the graph is (0, 4).
Let y = 0, then 3x = 20, x = 20/3 ≈ 6.67; so another point is (6.67, 0).
Plot these two points and draw a line passing through them.
For equation 2:
Let x = 0, then y = 16; hence, one point on the graph is (0, 16).
Let y = 0, then 3x = 16, x = 16/3 ≈ 5.33; so another point is (5.33, 0).
Plot these two points and draw a line passing through them.
For equation 3:
Let x = 0, then y = 2; hence, one point on the graph is (0, 2).
Let y = 0, then -2x = 2, x = -1; so another point is (-1, 0).
Plot these two points and draw a line passing through them.
Step 2: Identify the feasible region
The feasible region is the intersection of the shaded regions that satisfy all the inequalities. In this case, we have four inequalities as follows:
1. 3x + 5y ≥ 20
2. 3x + y ≤ 16
3. -2x + y ≤ 2
4. x ≥ 0, y ≥ 0
The feasible region is the area where all these conditions are satisfied. Shade the region where the inequalities hold true.
Step 3: Determine corner points
The corner points of the feasible region are the points where the inequality lines intersect. These points are the potential solutions to the linear programming problem.
Identify the intersection points of the lines. In this case, there are three intersection points: A, B, and C.
Step 4: Evaluate the objective function
Calculate the value of the objective function P = 3x + 2y at each of the corner points A, B, and C.
For point A:
x = 0, y = 4
P = 3(0) + 2(4) = 8
For point B:
x = 8/3, y = 16/3
P = 3(8/3) + 2(16/3) = 24/3 + 32/3 = 56/3 ≈ 18.67
For point C:
x = 2, y = 7
P = 3(2) + 2(7) = 6 + 14 = 20
Step 5: Determine min and max
The minimum and maximum values of P are found by comparing the values obtained at the corner points.
The minimum value: P = 8 at point A.
The maximum value: P = 20 at point C.
Therefore, the minimum value of P is 8 at the point (0, 4), and the maximum value of P is 20 at the point (2, 7).