Maximize p = 2x + 2y

subject to 4x + 8y 24
8x + 4y 32
x 0 , y 0
(Round all answers to the nearest 0.1.)
p = Incorrect: Your answer is incorrect.
x = Incorrect: Your answer is incorrect.
y = Incorrect: Your answer is incorrect.

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).

To solve this problem, we will use linear programming to find the maximum value of p given the given constraints.

1. Start by graphing the constraints on a coordinate plane.

The first constraint, 4x + 8y ≤ 24, can be rewritten as y ≤ 3 - 0.5x.
The second constraint, 8x + 4y ≤ 32, can be rewritten as y ≤ 8 - 2x.

Plot the lines y = 3 - 0.5x and y = 8 - 2x on a coordinate plane.

2. Determine the feasible region.

The feasible region is the area where the shaded regions for each constraint overlap. In this case, the feasible region is the portion of the coordinate plane below both lines.

3. Find the corner points of the feasible region.

To find the corner points, determine the coordinates where the lines intersect. In this case, the corner points are the points where the lines intersect and the axes.

The corner points in this case are:
A: (0,0)
B: (0,3)
C: (4,0)

4. Evaluate the objective function p at each corner point.

Simply substitute the coordinates of each corner point into the objective function p = 2x + 2y.

At corner point A: p = 2(0) + 2(0) = 0
At corner point B: p = 2(0) + 2(3) = 6
At corner point C: p = 2(4) + 2(0) = 8

5. Determine the maximum value of p.

Since p is a linear function, the maximum value occurs at one of the corner points. In this case, the maximum value of p is 8, which occurs at corner point C: (4,0).

Therefore, the maximum value of p is 8.

To summarize the answers to the question:
p = 8 (rounded to the nearest 0.1).
x = 4 (rounded to the nearest 0.1).
y = 0 (rounded to the nearest 0.1).