Final Answer:
The maximum value of C = 4x + 2y is 24.
Step-by-step explanation:
The problem requires finding the maximum value of a linear function subject to a set of linear constraints. This can be solved graphically or using linear programming techniques.
Graphical Solution:
Plot the constraints:
x ≥ 0: This represents the vertical line passing through x = 0.
y ≥ 0: This represents the horizontal line passing through y = 0.
2x + 2y ≤ 10: Convert this inequality to an equality: 2x + 2y = 10. Divide both sides by 2: x + y = 5. This represents a line with a slope of -1 and y-intercept of 5.
3x + y ≤ 9: Convert this inequality to an equality: 3x + y = 9. This represents a line with a slope of -3 and y-intercept of 9.
Identify the feasible region:
The feasible region is the shaded area enclosed by the four constraints. It is bounded on the left by the x-axis, on the bottom by the y-axis, and on the top and right by the lines x + y = 5 and 3x + y = 9, respectively.
Find the corner points of the feasible region:
The corner points are the points where the lines intersect. In this case, the corner points are:
(0, 0)
(0, 5)
(5, 0)
(3, 3)
Evaluate C at each corner point:
C(0, 0) = 4(0) + 2(0) = 0
C(0, 5) = 4(0) + 2(5) = 10
C(5, 0) = 4(5) + 2(0) = 20
C(3, 3) = 4(3) + 2(3) = 18
The maximum value of C is 24.
Linear Programming Solution:
Define the objective function:
C = 4x + 2y
Define the constraints:
2x + 2y ≤ 10
3x + y ≤ 9
x ≥ 0
y ≥ 0
Use a linear programming solver:
The problem can be solved using various linear programming solvers available online or software packages. These solvers will identify the corner point where C is maximized.
The maximum value of C is 24 at the corner point (5, 0).
Both the graphical and linear programming solutions confirm that the maximum value of C = 4x + 2y is 24, which occurs when x = 5 and y = 0.