Final answer:
To minimize c = 6x + 3y subject to the constraints, graphically solve the system of inequalities and find the coordinates of the vertices of the feasible region. Substitute these values into the objective function to find the minimum value of c.
Step-by-step explanation:
To minimize the objective function c = 6x + 3y subject to the constraints, we need to find the values of x and y that satisfy the constraints and minimize the value of c. Let's solve the system of inequalities:
a. 4x + 3y ≥ 24
b. 4x + y ≤ 0
c. y ≥ 0
To graphically solve this system, we first plot the lines representing the inequalities on a coordinate plane. The feasible region is the shaded area where all the inequalities are satisfied. We then find the coordinates of the vertices of this feasible region.
The vertices of the feasible region are (0, 8), (0, 0), and (6, 0). We substitute these values into the objective function to find the minimum value of c:
For (0, 8): c = 6(0) + 3(8) = 24
For (0, 0): c = 6(0) + 3(0) = 0
For (6, 0): c = 6(6) + 3(0) = 36
The minimum value of c is 0, which occurs at the point (0, 0).