SOLUTION: find the maximum value of p=5x+3y under the following constraints:2x+y=<20, 2x+3y=<24, x=>0 and y=>o

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Question 1086607: find the maximum value of p=5x+3y under the following constraints:2x+y=<20, 2x+3y=<24, x=>0 and y=>o

Answer by ikleyn(52832)   (Show Source): You can put this solution on YOUR website!
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find the maximum value of p=5x+3y under the following constraints:2x+y=<20, 2x+3y=<24, x=>0 and y=>o
~~~~~~~~~~~~~~~~~~~~~

I will solve it by the linear programming method.
I will not explain here how the method works in general case - find other sourses for it.
I will only explain how it works for the given concrete problem.

1.  Make a plot of the feasible area.
    It is restricted by the straight lines 
    
    2x +  y = 20,     or     y = 20 - 2x;
    2x + 3y = 24,     or     y = ;
    x = 0                    (y-axis)                and
    y = 0                    (x-axis).

    See the plot below:


              


         Feasibility area. Straight lines 2x + y = 20 (red);  2x + 3y = 24 (green)


    Feasibility area is this quadrilateral in the first quadrant restricted by the straight lines and axes.


2.  The critical points are the vertices of this quadrilateral.

    One vertex is x-intercept of the red line.      It is the point (10,0).

    Second vertex is y-intercept of the green line. It is the point (0,8).

    Third vertex is the intersection point of the red and the green line.
    To find the coordinates of this point, you need to solve the system of two linear equations.
    I did it for you and found the coordinates  (x,y) =             (9,2).


3.  The last step you need to do is to calculate the values of your "profit" function  P(x,y) = 5x + 3y at vertices. You have

    First vertex:   P(10,0) = 5*10 + 3*0 = 50.

    Second vertex:  P(0,8) = 5*0 + 3*8   = 24.

    Third vertex:   P(9,2) = 5*9 + 3*2 = 45 + 6 = 51.


4.  Your "profit" function is maximal at the third vertex.
   
    Then the linear programming method states that it is the solution of your minimax problem.


              The function p(x,y) = 5x + 3y has the maximum value over the feasibility area at the point (x,y) = (9,2), 
              and this maximum value is equal to 51.

The solution is completed. The problem is solved.


See also the lesson
    - Solving minimax problems by the Linear Programming method
in this site.



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