document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1137186>Question 1137186</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A farmer wants to see that his herd gets the minimum daily requirement of three basic nutrients A, B and C. Daily requirements are 14 for A, 12 for B and 18 for C. Product y1 has two units of A, and one unit each of B and C; product y2 has one unit each of A and B, and three units of C. The cost of y1 is $2 and the cost of y2 is $4.Determine the least-cost combination of y1 and y2 .<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #755077 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=755077\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x = the number of y1 products.<BR>\n" );
document.write( "let y = the number of y2 products.\r<BR>\n" );
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document.write( "product y1 has 2 units of A and 1 unit of B and 1 unit of C.<BR>\n" );
document.write( "product y2 has 1 unit of A and 1 unit of B and 3 units of C.\r<BR>\n" );
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document.write( "the total total requirement is for 14 units of A and 12 units of B and 18 units of C.\r<BR>\n" );
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document.write( "for the A requirement, your constraint equation is 2x + y >= 14<BR>\n" );
document.write( "for the B requirement, your constraint equation is x + y >= 12<BR>\n" );
document.write( "for the C requirement, your constraint equation is x + 3y >= 18\r<BR>\n" );
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document.write( "another requirement is that x can't be less than 0 and y can't be less than 0.\r<BR>\n" );
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document.write( "the inequality for that is x >= 0 and y >= 0.\r<BR>\n" );
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document.write( "your objective function is cost = 2 * x + 4 * y.\r<BR>\n" );
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document.write( "that's the function you want to minimize.\r<BR>\n" );
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document.write( "using the desmos.com calculator, you would graph the opposite of the inequalities.\r<BR>\n" );
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document.write( "the area of the graph that is not shaded will be your region of feasibility.\r<BR>\n" );
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document.write( "you will then evaluate your objective function at the corner points of this feasible region.\r<BR>\n" );
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document.write( "the corner point with the least cost and that satisfies the constraints is your solution.\r<BR>\n" );
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document.write( "the inequalities are:\r<BR>\n" );
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document.write( "2x + y >= 14<BR>\n" );
document.write( "x + y >= 12<BR>\n" );
document.write( "x + 3y >= 18<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "you will graph:\r<BR>\n" );
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document.write( "2x + y <= 14<BR>\n" );
document.write( "x + y <= 12<BR>\n" );
document.write( "x + 3y <= 18<BR>\n" );
document.write( "x <= 0<BR>\n" );
document.write( "y <= 0\r<BR>\n" );
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document.write( "the area of the graph that is not shaded is your region of feasibility.\r<BR>\n" );
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document.write( "here's that the graph looks like.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/032311.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the corner point of the feasible region and the evaluation of the objective function at those corner points is:\r<BR>\n" );
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document.write( "the corner points are (0,14), (2,10), (9,3), (18,0).\r<BR>\n" );
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document.write( "the objective function is evaluated at each corner point and the corner point with the least cost that satisfies the constraints is the solution.\r<BR>\n" );
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document.write( "that point is (9,3) where the total cost is 2 * 9 + 4 * 3 = 18 + 12 = 30.\r<BR>\n" );
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document.write( "you can check for yourself to see that this corner point has the least cost.\r<BR>\n" );
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document.write( "the constraints have to all be satisfied as well.\r<BR>\n" );
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document.write( "at the corner point of (9,3), .....\r<BR>\n" );
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document.write( "the number of units of A are 2 * 9 + 1 * 3 = 21 which is greater than or equal to 14.<BR>\n" );
document.write( "the number of units of B are 1 * 9 + 1 * 3 = 12 which is greater than or equal to 12.<BR>\n" );
document.write( "the number of units of C are 9 * 1 + 3 * 4 = 21 which is greater than or equal to 18.\r<BR>\n" );
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document.write( "the cost is minimum and all the constraints are satisfied, so you would buy 9 product y1 and 3 product y2 and your requirements will be satisfied and your cost will be minimum.\r<BR>\n" );
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