\r\n" );
document.write( "baking a tray of corn muffins takes 4 c milk and 3 c wheat flour. a tray of\r\n" );
document.write( "bran muffins takes 2 c milk and 3 c wheat flour.a baker has 16 c milk and 15 c\r\n" );
document.write( "wheat flour. he makes $3 profit per tray of corn muffins and $2 profit per tray\r\n" );
document.write( "of bran muffins. How many trays of each type of muffins should the baker make\r\n" );
document.write( "to maximize his profit.\r\n" );
document.write( "\r\n" );
document.write( "Let x = the number of trays of corn muffins\r\n" );
document.write( "Let y = the number of bran muffins\r\n" );
document.write( "\r\n" );
document.write( "First set up the milk inequality to make sure he\r\n" );
document.write( "doesn't run out of milk:\r\n" );
document.write( "\r\n" );
document.write( ">>...a tray of corn muffins takes 4 c milk...<<\r\n" );
document.write( "\r\n" );
document.write( "So x trays of corn muffins taks 4x c milk\r\n" );
document.write( "\r\n" );
document.write( ">>...a tray of bran muffins takes 2 c milk...<<\r\n" );
document.write( "\r\n" );
document.write( "So y trays of bran muffins taks 2y c milk\r\n" );
document.write( "\r\n" );
document.write( "So x trays of corn muffins and y trays of bran \r\n" );
document.write( " muffins takes 4x+2y c milk\r\n" );
document.write( "\r\n" );
document.write( ">>...a baker has 16 c milk...<<\r\n" );
document.write( "\r\n" );
document.write( "So to keep from running out of milk 4x+2y c milk \r\n" );
document.write( " must not be greater than 16 c, or\r\n" );
document.write( "\r\n" );
document.write( " 4x + 2y < 16\r\n" );
document.write( "\r\n" );
document.write( "--------------\r\n" );
document.write( "\r\n" );
document.write( "Now set up the wheat flour inequality to make sure he\r\n" );
document.write( "doesn't run out of wheat flour:\r\n" );
document.write( "\r\n" );
document.write( ">>...a tray of corn muffins takes...3 c wheat flour...<<\r\n" );
document.write( "\r\n" );
document.write( "So x trays of corn muffins taks 3x c wheat flour\r\n" );
document.write( "\r\n" );
document.write( ">>...a tray of bran muffins takes...3 c wheat flour...<<\r\n" );
document.write( "\r\n" );
document.write( "So y trays of bran muffins taks 3y c wheat flour\r\n" );
document.write( "\r\n" );
document.write( "So x trays of corn muffins and y trays of bran \r\n" );
document.write( " muffins takes 3x+3y c wheat flour\r\n" );
document.write( "\r\n" );
document.write( ">>...a baker has...15 c wheat flour...<<\r\n" );
document.write( "\r\n" );
document.write( "So to keep from running out of wheat flour, 3x+3y c milk \r\n" );
document.write( " must not be greater than 15 c, or\r\n" );
document.write( "\r\n" );
document.write( " 3x + 3y < 15 \r\n" );
document.write( "\r\n" );
document.write( "-----------------------------\r\n" );
document.write( "\r\n" );
document.write( "Now there are two obvious inequalities which state that\r\n" );
document.write( "the number of c of milk and the number of c of wheat\r\n" );
document.write( "flour must not be negative:\r\n" );
document.write( "\r\n" );
document.write( " x > 0\r\n" );
document.write( " y > 0\r\n" );
document.write( "\r\n" );
document.write( "So our set of inequalities are\r\n" );
document.write( "\r\n" );
document.write( " 4x + 2y < 16\r\n" );
document.write( " 3x + 3y < 15\r\n" );
document.write( " x > 0\r\n" );
document.write( " y > 0\r\n" );
document.write( "\r\n" );
document.write( "Now we grapg the boundary lines, whose equations are \r\n" );
document.write( "gotten by replacing the symbols of inequality by equal\r\n" );
document.write( "signs:\r\n" );
document.write( "\r\n" );
document.write( " 4x + 2y = 16 [whose intercepts are (0,8) and (4,0) \r\n" );
document.write( " 3x + 3y = 15 (whose intercepts are (0,5) and (5,0)\r\n" );
document.write( " x = 0 (the y-axis)\r\n" );
document.write( " y = 0 (the x-axis)\r\n" );
document.write( " \r\n" );
document.write( "Now we graph those lines. We only need the upper right hand \r\n" );
document.write( "side of the xy-plane since neither x nor y can be negative:\r\n" );
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document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "We shade the feasible region which is below the red and green \r\n" );
document.write( "lines which is above the x-axis and to the right of the y-axis. \r\n" );
document.write( "I can't shade that region here but you can on your paper. I \r\n" );
document.write( "will chop off the lines where they intersect. We do need to\r\n" );
document.write( "solve the two slanted lines (the red and green ones to find\r\n" );
document.write( "the point where they intersect:\r\n" );
document.write( "\r\n" );
document.write( " 4x + 2y = 16\r\n" );
document.write( " 3x + 3y = 15\r\n" );
document.write( "\r\n" );
document.write( "They have solution (x,y) = (3,2), so I will chop off the \r\n" );
document.write( "lines at that point, and at the axes.\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Now this is the feasible region polygon. It has corner points\r\n" );
document.write( "(0,0), (0,5), (3,2), and (4,0)\r\n" );
document.write( "\r\n" );
document.write( "Now we will set up the profit equation:\r\n" );
document.write( "\r\n" );
document.write( ">>...he makes $3 profit per tray of corn muffins...<<\r\n" );
document.write( "\r\n" );
document.write( "So his profit for x trays of corn muffins is $3x \r\n" );
document.write( "\r\n" );
document.write( ">>,,,he makes...$2 profit per tray of bran muffins...<<\r\n" );
document.write( "\r\n" );
document.write( "So his profit for y trays of bran muffins is $2y\r\n" );
document.write( "\r\n" );
document.write( "So his total profit P is $3x + $2y and the profit \r\n" );
document.write( "equation is\r\n" );
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document.write( " P = 3x + 2y\r\n" );
document.write( "\r\n" );
document.write( "Both his maximimum profit (and his minimum profit)\r\n" );
document.write( "must be at one of the four corner points:\r\n" );
document.write( "\r\n" );
document.write( "Corner point | 3x + 2y = P \r\n" );
document.write( " (0,0) | 3(0) + 2(0) = $0\r\n" );
document.write( " (0,5) | 3(0) + 2(5) = $10 \r\n" );
document.write( " (3,2) | 3(3) + 2(2) = $13\r\n" );
document.write( " (4,0) | 4(4) + 2(0) = $16\r\n" );
document.write( "\r\n" );
document.write( "The corner point (0,0) represents his minimum\r\n" );
document.write( "profit where he doesn't make any muffins at all\r\n" );
document.write( "and doesn't make any money.\r\n" );
document.write( "\r\n" );
document.write( "The corner point (0,5) represents the case when\r\n" );
document.write( "he makes no corn muffins and 5 bran trays of bran\r\n" );
document.write( "muffins for a profit of $10.\r\n" );
document.write( "\r\n" );
document.write( "The corner point (3,2) represents the case when\r\n" );
document.write( "he makes 3 trays of corn muffins and 2 trays of\r\n" );
document.write( "bran muffins trays of bran muffins for a profit \r\n" );
document.write( "of $13.\r\n" );
document.write( "\r\n" );
document.write( "The corner point (4,0) represents the case when\r\n" );
document.write( "he makes 4 trays of corn muffins and no bran \r\n" );
document.write( "muffins for a profit of $16.\r\n" );
document.write( " \r\n" );
document.write( "So he maximizes his profit when he makes 4 trays\r\n" );
document.write( "of corn muffins an no bran muffins for a profit \r\n" );
document.write( "of $16.\r\n" );
document.write( "\r\n" );
document.write( "--------------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "Note that if the profit on the two kinds of\r\n" );
document.write( "muffins had been reversed, so that he made\r\n" );
document.write( "$2 profit on the corn muffin trays and $3\r\n" );
document.write( "profit on the bran muffins, the profit\r\n" );
document.write( "equation would have been P = 2x + 3y\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Corner point | 2x + 3y = P \r\n" );
document.write( " (0,0) | 2(0) + 3(0) = $0\r\n" );
document.write( " (0,5) | 2(0) + 3(5) = $15 \r\n" );
document.write( " (3,2) | 2(3) + 3(2) = $12\r\n" );
document.write( " (4,0) | 2(4) + 3(0) = $8\r\n" );
document.write( "\r\n" );
document.write( "the maximum profit would have been \r\n" );
document.write( "reporesented by the corner point (0,5)\r\n" );
document.write( "and he would have made 5 trays of bran \r\n" );
document.write( "muffins and no corn muffins.\r\n" );
document.write( "\r\n" );
document.write( "--------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "But notice that if his profit had been $2.75\r\n" );
document.write( "per corn muffin tray snd $2.25 per bran\r\n" );
document.write( "muffin tray, the profit equation would\r\n" );
document.write( "have been P = 2.25x + 2.75y \r\n" );
document.write( "\r\n" );
document.write( "Corner point | 2.75x + 2.25y = P \r\n" );
document.write( " (0,0) | 2.75(0) + 2.25(0) = $0\r\n" );
document.write( " (0,5) | 2.75(0) + 2.25(5) = $11.25 \r\n" );
document.write( " (3,2) | 2.75(3) + 2.25(2) = $12.75\r\n" );
document.write( " (4,0) | 2.75(4) + 2.25(0) = $11.00\r\n" );
document.write( "\r\n" );
document.write( "the maximum profit would have been represented\r\n" );
document.write( "by the corner point (3,2) which would have\r\n" );
document.write( "been the case where he made 3 trays of corn\r\n" );
document.write( "muffins and 2 trays of bran muffins. \r\n" );
document.write( "and no corn muffins.\r\n" );
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document.write( "So, as you see, which corner point represents\r\n" );
document.write( "the maximum profit depends on the profit for\r\n" );
document.write( "each item. If there had been a loss on each\r\n" );
document.write( "kind of muffin, then the maximum profit would\r\n" );
document.write( "have been $0 so instead of taking a loss, he\r\n" );
document.write( "would have not made any muffins at all and the\r\n" );
document.write( "corner point (0.0) would have been the wisest\r\n" );
document.write( "decision. \r\n" );
document.write( "\r\n" );
document.write( "Edwin McCravy
\n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
