Linear Programming

A zoologist conducts two types of experiments with lizards.  An experiment of type 1 requires three brown lizards and one gray lizard, whereas an experiment of type 2 requires one brown lizard and two gray lizards.  Each experiment of type 1 earns one point while each experiment of type 2 earns two points.  There are 60 brown lizards and 40 gray lizards available.  How many experiments of each type should be performed to mazimize the total points earned?
 

I.  Decide what x and y will represent:

II.  Write an expression for the object being maximized or minimized:

     P = x + 2y
 
 

III.  Write inequalities representing the constraints expressed in the problem:

    3x + y £ 60
    x + 2y £ 40
    x  ³ 0
    y  ³ 0
 

IV.  Graph the equations that belong to these inequalities:

  The intersection of the shaded areas forms a quadrilateral.
           This picture is the part of the graphs in the first quadrant since x and y are both greater
           than 0.
 
 

V.  Find the vertices of the quadrilateral.

VI.  Substitute each one of these points into the "points" expression and find the one 
        that gives the maximum value of points.

      P = x + 2y
      P = 0 + 2(20) = 40
      P = 20 + 2(0) = 20
      P = 16 + 2(12) = 16 + 24 = 40

      40 is the maximum number of points.  It occurs when the zoologist does no type 1
      experiments and 20 type 2 experiments; and it occurs when the zoologist does 16 type 1
      experiments and 12 type 2 experiments.  Since he wants to conduct two types of
      experiments, we will use (16, 12).