document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=904149>Question 904149</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A farming cooperative mixes two brands of cattle feed. Brand X costs $25 per bag and contains 2 units of nutritional element A, 2 units of element B, and 2 units of element C. Brand Y costs $20 per bag and contains 1 unit of nutritional element A, 9 units of element B, and 3 units of element C. The minimum requirements for nutrients A,B,& C are 12 units, 36 units, and 24 units, respectively. Find the number of bags of each brand that should be mixed to produce a mixture having a minimum cost per bag. What is the minimum cost per bag?\r<BR>\n" );
document.write( "\n" );
document.write( "Please show all your work. I am really bad with Linear Programming.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #548566 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=548566\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x equal the number of bags of Brand X.<BR>\n" );
document.write( "let y equal the number of bags of Brand Y.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The total cost will be 25*x + 20*y<BR>\n" );
document.write( "That's your objective function that you want to minimize.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your constraints are given in the following table:<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Nutritional Element      Brand X     Brand Y\r\n" );
document.write( "A                           2           1\r\n" );
document.write( "B                           2           9\r\n" );
document.write( "C                           2           3\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A >= 12 units<BR>\n" );
document.write( "B >= 36 units<BR>\n" );
document.write( "C >= 24 units\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Your constraint equations are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2x + y >= 12 units of nutritional element A<BR>\n" );
document.write( "2x + 9y >= 36 units of nutritional element B<BR>\n" );
document.write( "2x + 3y >= 24 units of nutritional element C\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are two other constraints that are implied and also need to be considered.<BR>\n" );
document.write( "They are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This is because the number of bags can never be less than 0.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You need to graph your constraints and then find the feasible region that satisfies all the constraints and then find the corners of the feasible region.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Your minimum cost will be at these corners.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You will graph the equality portion of the constraints.<BR>\n" );
document.write( "Therefore you will graph:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2x + y = 12 units of nutritional element A<BR>\n" );
document.write( "2x + 9y = 36 units of nutritional element B<BR>\n" );
document.write( "2x + 3y = 24 units of nutritional element C<BR>\n" );
document.write( "x = 0<BR>\n" );
document.write( "y = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Once you do that, you will find the area of the graph that satisfies all the inequality constraints.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We'll do the graph first and then find the feasible region.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The graph will look like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/092102.jpg\" alt = \"$$$\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now you look for the feasible region.<BR>\n" );
document.write( "The feasible region is the area of the graph that satisfies all the inequality equations.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The feasible area will be the area on the graph that is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "on or above the line y = 12 - 2x<BR>\n" );
document.write( "on or above the line y = (36-2x)/9<BR>\n" );
document.write( "on or above the line y = (24-2x)/3<BR>\n" );
document.write( "on or above the line y = 0<BR>\n" );
document.write( "on or to the right of the line x = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you would have to look at all the lines and determine in which direction the area of feasibility lies.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I could do it manually but it will take too much time and a lot of effort to graph it manually and why do that when I have software to do it for me.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I will show you where the area is so you can see it.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You should look to determine what the area is manually from the first graph I showed you.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The feasible region is the area on the graph that is the darkest shade of gray.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You can see that it is the area that is above all of the constraint lines on the graph.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The maximum / minimum solution will be at the corners of that feasible region which are shown on the graph with their coordinate points.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/092103.jpg\" alt = \"$$$\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "What's left is to find the minimum cost solution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "That will be at the coordinate points of:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "   x     y     25x + 20y\r\n" );
document.write( "\r\n" );
document.write( "   0     12    25*0 + 20*12 = 0 + 240 = 240\r\n" );
document.write( "   3     6     25*3 + 20*6 = 75 + 120 = 195 *****\r\n" );
document.write( "   9     2     25*9 + 20*2 = 225 + 40 = 265\r\n" );
document.write( "   18    0     25*18 + 20*0 = 450 + 0 = 450\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The minimum cost is when he makes 3 bags of brand X and 6 bags of brand Y.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "All the constraints have to be met as well.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the constraint equations are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2x + y >= 12 units of nutritional element A<BR>\n" );
document.write( "2x + 9y >= 36 units of nutritional element B<BR>\n" );
document.write( "2x + 3y >= 24 units of nutritional element C<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "When x = 3 and y = 6:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2x + y becomes 2*3 + 1*6 = 6 + 6 = 12 so the first constraint is met.<BR>\n" );
document.write( "2x + 9y becomes 2*3 + 9*6 = 6 + 54 = 60 so the second constraint is met.<BR>\n" );
document.write( "2x + 3y becomes 2*3 + 3*6 = 6 + 18 = 24 so the third constraint is met.<BR>\n" );
document.write( "x and y are both greater than 0 so the fourth and fifth constraints are also met.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Looks like the solution is valid.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The minimum cost is when he produces 3 bags of Brand X and 6 bags of brand Y at a total cost of $195.00.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "    </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );