To solve this linear programming problem and maximize the objective function p=2x+2y subject to the given constraints, we can follow these steps:
Step 1: Graph the constraints:
Plot the lines 4x + 8y = 24 and 8x + 4y = 32 on a coordinate plane. To do this, we can find the x and y-intercepts for each line by setting x and y equal to zero.
For the first constraint:
When x = 0, we have 4(0) + 8y = 24, which gives us y = 3. So we plot the point (0, 3).
When y = 0, we have 4x + 8(0) = 24, which gives us x = 6. So we plot the point (6, 0).
Connect these two points to form a line.
For the second constraint:
When x = 0, we have 8(0) + 4y = 32, which gives us y = 8. So we plot the point (0, 8).
When y = 0, we have 8x + 4(0) = 32, which gives us x = 4. So we plot the point (4, 0).
Connect these two points to form a line.
Step 2: Identify the feasible region:
The feasible region is the region where both constraints are satisfied. In this case, it is the region where the two lines intersect or overlap.
Step 3: Determine the corner points of the feasible region:
To find the corner points of the feasible region, we can solve the two equations simultaneously:
4x + 8y = 24
8x + 4y = 32
Divide the first equation by 4, we get:
x + 2y = 6
Now subtract 2 times this equation from the second equation, we get:
8x + 4y - 4(x + 2y) = 32 - 4(6)
8x + 4y - 4x - 8y = 32 - 24
4x - 4y = 8
Simplify this equation:
x - y = 2
Now we can solve the system of equations:
x + 2y = 6
x - y = 2
Adding the two equations, we get:
2x + y = 8
Subtracting the second equation from the first, we get:
3y = 4
Divide both sides by 3, we get:
y = 4/3
Substitute this value of y into either of the original equations, we get:
x + 2(4/3) = 6
x + 8/3 = 6
x = 6 - 8/3
x = (18 - 8)/3
x = 10/3
So the first corner point is (10/3, 4/3).
Step 4: Evaluate the objective function at each corner point:
Now we can substitute each of the corner points into the objective function p=2x+2y and evaluate the result:
For the first corner point (10/3, 4/3):
p = 2(10/3) + 2(4/3)
p = 20/3 + 8/3
p = 28/3 = 9.3
Therefore, the maximum value of p is 9.3 at the corner point (10/3, 4/3).