SOLUTION: Maximize:Z=40x+88y Subject to:2x+8y≤60 5x+2y≥60,x,y,>0

Algebra.Com
Question 1104488: Maximize:Z=40x+88y
Subject to:2x+8y≤60
5x+2y≥60,x,y,>0
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52787)   (Show Source): You can put this solution on YOUR website!
.
Read the lesson
    - Solving minimax problems by the Linear Programming method,  Problem 2   and   Problem 3.
in this site.

You will find all necessary instructions there, presented for another similar problem.


Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


The constraints are equivalent to
and

Here is a graph; the feasibility region is below the red line and above the green line. The vertices of the feasibility region are (12,0), (30,0), and (10,5).



The standard method for finding the maximum value of the objective function is to evaluate that function at each vertex of the reasibility region:

(12,0): 40x+88y = 40(12) = 480
(30,0): 40x+88y = 40(30) = 1200
(10,5): 40x+88y = 40(10)+88(5) = 400+440 = 840

The maximum value is 1200 at (30,0).

In fact there is no need to evaluate the objective function at every vertex of the feasibility region. You can tell which vertex will produce the maximum value of the objective function by comparing the slope of the objective function with the slopes of the constraint lines.

The slope of the objective function is -40/88 = -5/11. So consider all the lines with slope -5/11:
A line with slope -5/11 passing through (12,0) will just touch a vertex;
a line with slope -5/11 passing through (5,10) will cut through the feasibility region;
a line with slope -5/11 passing through (30,0) will again touch the feasibility region just at a vertex.

We can conclude from that -- without evaluating the objective function at any of the three vertices -- that the minimum value will be at (12,0), the maximum value will be at (30,0), and an intermediate value will occur at (10,5).

Then, since the problem asks for the maximum value of the objective function, we only need to evaluate it at (30,0).
RELATED QUESTIONS

Maximize z=8x+12y subject to 40x+80y≤​560, 6x+8y≤​72,... (answered by greenestamps)
Maximize p = x + y subject to x + 2y ≥ 30 2x + 2y ≤ 30 2x + y ≥ 30... (answered by ikleyn)
Maximize z = 2x + 4y subject to the constraints x ≥ 0, y ≥... (answered by Edwin McCravy)
Maximize z = 2x + 4y subject to the constraints x ≥ 0, y ≥ 0, x + y (answered by solver91311)
Use the simplex method to maximize p= 5x+4y+3z subject to the constraints: x>=0, y>=0,... (answered by Edwin McCravy)
Maximize P = 3x + 2y
subject to
  2x + y ≤ 18
2x + 3y  ≤ 42
 3x + y   (answered by Edwin McCravy)

Solve the LP problem. If no optimal solution exists, indicate whether the feasible region  (answered by greenestamps,ikleyn)

Maximize:z=40x+12y

Subject to:
6x-2y\le 9
3x+y\le 9
x\ge 0
y\ge... (answered by Theo)

maximize z=5x+2y subject to... (answered by ikleyn)