Find the maximum value of the objective function and where it occurs, subject to the indicated constraints.
Objective function:

Z = 5x + 8y

Constraints:

X ≥ 0
Y ≥ 0
X + 3y ≤ 15
4x + y ≤ 16

A)Maximum at (3, 4): 47
B)Maximum at (4, 0): 20
C)Maximum at (5, 0): 44
D)Maximum at (0, 5): 40
E)Maximum at (0, 0): 0

Question

Find the maximum value of the objective function and where it occurs, subject to the indicated constraints. ​
Objective function:
​
​
Z = 5x + 8y
​
Constraints:
​
X ≥ 0
Y ≥ 0
X + 3y ≤ 15
4x + y ≤ 16
​​  ​
A)Maximum at (3, 4): 47 
B)Maximum at (4, 0): 20 
C)Maximum at (5, 0): 44 
D)Maximum at (0, 5): 40 
E)Maximum at (0, 0): 0

Quizplus · Accepted Answer

We can solve this problem using graphical or algebraic method. 
Graphical method: Plotting the constraints on the coordinate axis and finding the region of feasible solutions. Then, computing the value of the objective function at the vertices of the feasible region and choosing the maximum.
 
By solving the two equations, we get the intersection points:

x = 3, y = 4  and x = 4, y = 1

(0,5) and (5,0) are on the edges but not the vertices. So, we don't need to evaluate them.

Next, we calculate the value of the objective function at each vertex: 
Z(0,0) = 0, Z(3,4) = 47, Z(4,1) = 37

Hence, the maximum value of the objective function is 47 and it occurs at (3, 4).

Find the Maximum Value of the Objective Function and Where