The feasible region is determined by the system of constraints.
First, let's graph the constraints to find the vertices of the feasible region.
Constraint 1: A + B ≥ 3
This equation represents a line in the coordinate plane. To graph it, we can find two points on the line:
When A = 0, B = 3
When B = 0, A = 3
Plotting these points and drawing the line through them, we have:
*[illustration 1.png]
Constraint 2: 204 + 10B ≤ 80
Simplifying this inequality, we have:
10B ≤ -124
B ≤ -12.4
Since B cannot be negative (as it represents the number of books), this constraint is not feasible.
Constraint 3: B ≤ 2A
This constraint can be graphed similarly to Constraint 1. Finding two points on the line:
When A = 0, B = 0
When A = 3, B = 6
Plotting these points and drawing the line through them, we have:
*[illustration 2.png]
The feasible region is the shaded region where the two lines intersect:
*[illustration 3.png]
Now, let's find the vertices of the feasible region:
Looking at the graph, we can see the vertices are:
(0, 3)
(3, 6)
Now, let's find the maximum value of the objective function within this feasible region.
To do this, we substitute the coordinates of each vertex into the objective function and find the maximum value.
For the vertex (0, 3):
G = 24 + 3(3)
G = 24 + 9
G = 33
For the vertex (3, 6):
G = 24 + 3(6)
G = 24 + 18
G = 42
The maximum value of the objective function G = 24 + 3B within the feasible region is 42, which occurs at the vertex (3, 6).
Given the system of constraints, name all vertices of the feasible region. Then find the maximum value of the given objective function.
OBJECTIVE: G = 24 + 3B
CONSTRAINTS:
A + B ≥ 3
204 + 10B ≤ 80
B ≤ 2A
1 answer