baking a tray of corn muffins takes 4 c milk and 3 c wheat flour. a tray of
bran muffins takes 2 c milk and 3 c wheat flour.a baker has 16 c milk and 15 c
wheat flour. he makes $3 profit per tray of corn muffins and $2 profit per tray
of bran muffins. How many trays of each type of muffins should the baker make
to maximize his profit.

Let x = the number of trays of corn muffins
Let y = the number of bran muffins

First set up the milk inequality to make sure he
doesn't run out of milk:

>>...a tray of corn muffins takes 4 c milk...<<

So x trays of corn muffins taks 4x c milk

>>...a tray of bran muffins takes 2 c milk...<<

So y trays of bran muffins taks 2y c milk

So x trays of corn muffins and y trays of bran 
     muffins takes 4x+2y c milk

>>...a baker has 16 c milk...<<

So to keep from running out of milk 4x+2y c milk 
     must not be greater than 16 c, or

          4x + 2y < 16

--------------

Now set up the wheat flour inequality to make sure he
doesn't run out of wheat flour:

>>...a tray of corn muffins takes...3 c wheat flour...<<

So x trays of corn muffins taks 3x c wheat flour

>>...a tray of bran muffins takes...3 c wheat flour...<<

So y trays of bran muffins taks 3y c wheat flour

So x trays of corn muffins and y trays of bran 
     muffins takes 3x+3y c wheat flour

>>...a baker has...15 c wheat flour...<<

So to keep from running out of wheat flour, 3x+3y c milk 
     must not be greater than 15 c, or

          3x + 3y < 15 

-----------------------------

Now there are two obvious inequalities which state that
the number of c of milk and the number of c of wheat
flour must not be negative:

                x > 0
                y > 0

So our set of inequalities are

          4x + 2y < 16
          3x + 3y < 15
                x > 0
                y > 0

Now we grapg the boundary lines, whose equations are 
gotten by replacing the symbols of inequality by equal
signs:

          4x + 2y = 16 [whose intercepts are (0,8) and (4,0)   
          3x + 3y = 15 (whose intercepts are (0,5) and (5,0)
                x = 0  (the y-axis)
                y = 0  (the x-axis)
 
Now we graph those lines. We only need the upper right hand 
side of the xy-plane since neither x nor y can be negative:



We shade the feasible region which is below the red and green 
lines which is above the x-axis and to the right of the y-axis. 
I can't shade that region here but you can on your paper. I 
will chop off the lines where they intersect.  We do need to
solve the two slanted lines (the red and green ones to find
the point where they intersect:

          4x + 2y = 16
          3x + 3y = 15

They have solution (x,y) = (3,2), so I will chop off the 
lines at that point, and at the axes.



Now this is the feasible region polygon.  It has corner points
(0,0), (0,5), (3,2), and (4,0)

Now we will set up the profit equation:

>>...he makes $3 profit per tray of corn muffins...<<

So his profit for x trays of corn muffins is $3x 

>>,,,he makes...$2 profit per tray of bran muffins...<<

So his profit for y trays of bran muffins is $2y

So his total profit P is $3x + $2y and the profit 
equation is

             P = 3x + 2y

Both his maximimum profit (and his minimum profit)
must be at one of the four corner points:

Corner point |  3x + 2y    =  P 
  (0,0)      | 3(0) + 2(0) =  $0
  (0,5)      | 3(0) + 2(5) = $10 
  (3,2)      | 3(3) + 2(2) = $13
  (4,0)      | 4(4) + 2(0) = $16

The corner point (0,0) represents his minimum
profit where he doesn't make any muffins at all
and doesn't make any money.

The corner point (0,5) represents the case when
he makes no corn muffins and 5 bran trays of bran
muffins for a profit of $10.

The corner point (3,2) represents the case when
he makes 3 trays of corn muffins and 2 trays of
bran muffins trays of bran muffins for a profit 
of $13.

The corner point (4,0) represents the case when
he makes 4 trays of corn muffins and no bran 
muffins for a profit of $16.
    
So he maximizes his profit when he makes 4 trays
of corn muffins an no bran muffins for a profit 
of $16.

--------------------------------------------

Note that if the profit on the two kinds of
muffins had been reversed, so that he made
$2 profit on the corn muffin trays and $3
profit on the bran muffins, the profit
equation would have been P = 2x + 3y


Corner point |  2x + 3y    =  P 
  (0,0)      | 2(0) + 3(0) =  $0
  (0,5)      | 2(0) + 3(5) = $15 
  (3,2)      | 2(3) + 3(2) = $12
  (4,0)      | 2(4) + 3(0) = $8

the maximum profit would have been 
reporesented by the corner point (0,5)
and he would have made 5 trays of bran 
muffins and no corn muffins.

--------------------------------------

But notice that if his profit had been $2.75
per corn muffin tray snd $2.25 per bran
muffin tray, the profit equation would
have been P = 2.25x + 2.75y 

Corner point |  2.75x + 2.25y    =  P 
  (0,0)      | 2.75(0) + 2.25(0) =  $0
  (0,5)      | 2.75(0) + 2.25(5) = $11.25 
  (3,2)      | 2.75(3) + 2.25(2) = $12.75
  (4,0)      | 2.75(4) + 2.25(0) = $11.00

the maximum profit would have been represented
by the corner point (3,2) which would have
been the case where he made 3 trays of corn
muffins and 2 trays of bran muffins. 
and no corn muffins.

So, as you see, which corner point represents
the maximum profit depends on the profit for
each item. If there had been a loss on each
kind of muffin, then the maximum profit would
have been $0 so instead of taking a loss, he
would have not made any muffins at all and the
corner point (0.0) would have been the wisest
decision.  

Edwin McCravy