Question

Find the maximum value of P = 7x + 9y

subject to the following constraints: {2x + 8y ≤ 64 {3x + 9y ≥ 81 {x ≥ 0 {y ≥ 0

Answer 1

To find the maximum value of P = 7x + 9y subject to the given constraints, you can use linear programming techniques. Here's the solution step by step:

1. Start by graphing the inequalities to identify the feasible region:

a. 2x + 8y ≤ 64 can be simplified to x + 4y ≤ 32.
b. 3x + 9y ≥ 81 can be simplified to x + 3y ≥ 27.

Now, graph the lines x + 4y = 32 and x + 3y = 27. The feasible region is the area where these lines intersect and satisfies the constraints.

2. Next, determine the vertices of the feasible region by finding the points of intersection of the lines:

a. Solve x + 4y = 32 and x + 3y = 27 simultaneously to find the point of intersection, which is (8, 6).

3. Calculate the value of P at each vertex:

a. P1 = 7(0) + 9(0) = 0 at the origin (0, 0).
b. P2 = 7(8) + 9(6) = 56 + 54 = 110 at the point (8, 6).

4. Compare the values of P at the vertices:

P1 = 0
P2 = 110

The maximum value of P is achieved at the vertex (8, 6), which is P = 110.

So, the maximum value of P subject to the given constraints is 110, and it occurs when x = 8 and y = 6.

Answer 2

Answer: To find the maximum value of P = 7x + 9y, we need to optimize the objective function P while satisfying the given constraints.

Let's analyze the constraints:

1. 2x + 8y ≤ 64: This constraint represents an upper bound on the combination of x and y values. It means that the sum of 2 times x and 8 times y must be less than or equal to 64.

2. 3x + 9y ≥ 81: This constraint represents a lower bound on the combination of x and y values. It means that the sum of 3 times x and 9 times y must be greater than or equal to 81.

3. x ≥ 0: This constraint specifies that x must be greater than or equal to 0, which means x cannot be negative.

4. y ≥ 0: This constraint specifies that y must be greater than or equal to 0, which means y cannot be negative.

To solve this problem, we can use the method of linear programming. We can graphically represent the constraints on a coordinate plane and find the feasible region where all the constraints are satisfied. Then, we can find the corner points of the feasible region and evaluate the objective function at those points to determine the maximum value of P.

However, since we cannot provide visual aids here, I will solve the problem algebraically.

To find the maximum value of P, we need to find the corner points of the feasible region.

Step 1: Graph the inequalities on a coordinate plane to find the feasible region.

Step 2: Identify the corner points of the feasible region. These corner points are the intersections of the lines representing the constraints.

Step 3: Evaluate the objective function P = 7x + 9y at each corner point.

The corner points of the feasible region are the solutions to the system of equations formed by the constraints.

By solving the system of equations, we find the following corner points:

Corner point 1: (0, 9)

Corner point 2: (16, 0)

Corner point 3: (32, 0)

Step 4: Evaluate the objective function at each corner point.

P1 = 7(0) + 9(9) = 0 + 81 = 81

P2 = 7(16) + 9(0) = 112 + 0 = 112

P3 = 7(32) + 9(0) = 224 + 0 = 224

Step 5: Compare the values of P at each corner point and determine the maximum value.

The maximum value of P is 224, which occurs at the corner point (32, 0).

Therefore, the maximum value of P = 7x + 9y subject to the given constraints is 224 when x = 32 and y = 0.