SOLUTION: Carlo and Anita make mailboxes and toys in their craft shop near Lincoln. Each mailbox requires 4 hours of work from Carlo and 4hours from Anita. Each toy requires 2 hours of work

The question is: how many mailboxes (X) and how many toys (Y) should be produced to maximize the revenue $11*X + $21*Y
under these restrictions:

4X + 2Y <= 24     (1)     (Carlo restricted by 24 hours per week) and
4X + 4Y <= 32     (2)     (Anita restricted by 32 hours per week).

In other words, you must maximize the objective function (revenue) 

R(X,Y) = 13X + 21Y

over the domain on the plot below, which is  a quadrilateral in the first quadrant (X >= 0,  Y >= 0) restricted 
by the red and the green lines.





Plots y =   (red) and y =  (green)



The method of linear programming says:

    1) Take the vertices of this quadrilateral

        (x1,Y1) = (0,8)   (green line Y-intercept)
        (x2,Y2) = (6,0)   (red line X-intercept)
        (x3,Y3) = (4,4)   (intersection point of the straight lines Y =  and Y =  )

    2) Calculate the objective function at these points

        R(X1,Y1) = 13*0 + 21*8 = 168;

        R(X2,Y2) = 13*6 + 21*0 = 78;

        R(X3,Y3) = 13*4 + 21*4 = 136.


    3) Then select one of these point where the objective function is maximal - In our case this point is (X1,Y1) = (0,8)


    4) This point gives your optimal solution X = 0 mailboxes and Y = 8 toys.


If they follow this optimal solution, their weekly revenue will be MAXIMAL, $168.