mamixmize P= 5x + 5y
Subject to 2x + y < 10
x + 2y < 8
x,y >10
To find the maximum value of the objective function P = 5x + 5y, subject to the given constraints, you can use the method of linear programming. This technique involves graphing the feasible region determined by the constraints and identifying the corner points of that region. Then, you evaluate the objective function at each corner point to find the maximum value of P.
Step 1: Graph the constraints:
Start by graphing the inequalities 2x + y < 10 and x + 2y < 8.
For the first constraint: 2x + y < 10
Rearrange the inequality to isolate y:
y < 10 - 2x
Plot the boundary line: 2x + y = 10
To do this, find two points that lie on the line by choosing values of x and solving for y. For example, when x = 0, y = 10, and when x = 5, y = 0. Connect these two points with a straight line.
For the second constraint: x + 2y < 8
Rearrange the inequality to isolate y:
y < (8 - x)/2
Plot the boundary line: x + 2y = 8
Again, find two points that lie on the line, such as when x = 0, y = 4, and when x = 8, y = 0. Connect these two points with a straight line.
Step 2: Identify the feasible region:
Shade the region that satisfies both inequalities. Since the constraints are strict inequalities (<), shade the region below both boundary lines.
Step 3: Find the corner points:
The corners or vertices of the feasible region are the intersection points of the two boundary lines. In this case, it should form a triangle with three corner points.
Step 4: Evaluate the objective function at each corner point:
Using each corner point's coordinates (x, y), substitute them into the objective function P = 5x + 5y. Calculate the value of P for each point.
Step 5: Determine the maximum value of P:
Compare the values of P obtained for all corner points. The corner point with the highest value of P will give you the maximum value of the objective function that satisfies the given constraints.