document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189015>Question 1189015</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A small petroleum company owns two reneries. Renery 1 costs $20,000 per day to operate, and it can produce<BR>\n" );
document.write( "400 barrels of high-grade oil, 300 barrels of medium-grade oil, and 200 barrels of low-grade oil each day. Renery<BR>\n" );
document.write( "2 is newer and more modern. It costs $25,000 per day to operate, and it can produce 300 barrels of high-grade<BR>\n" );
document.write( "oil, 400 barrels of medium-grade oil, and 500 barrels of low-grade oil each day. The company has orders totaling<BR>\n" );
document.write( "25,000 barrels of high-grade oil, 27,000 barrels of medium-grade oil, and 30,000 barrels of low-grade oil. How<BR>\n" );
document.write( "many days should it run each renery to minimize its costs and still rene enough oil to meet its orders? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #820233 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=820233\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your objective function is 20,000 * x + 25,000 * y = cost<BR>\n" );
document.write( "this is what you want to minimize.\r<BR>\n" );
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document.write( "your constraint functions are:\r<BR>\n" );
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document.write( "400x * x + 300 * y >= 25,000<BR>\n" );
document.write( "300x + 400y >= 27,000<BR>\n" );
document.write( "200x + 500y >= 30,000<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "using the desmos.com calculator at <a  rel=nofollow HREF= https://www.desmos.com/calculator\"\" target = \"_blank\">https://www.desmos.com/calculator</a>, you would do the following:\r<BR>\n" );
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document.write( "graph the opposite of the inequalities.\r<BR>\n" );
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document.write( "the feasible region is the area on the graph  that is not shaded.<BR>\n" );
document.write( "evaluate the objective function at each of the corner points.<BR>\n" );
document.write( "the minimum cost will be at one of those corner pointas.\r<BR>\n" );
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document.write( "the graph is shown below.<BR>\n" );
document.write( "the corner points are in (x,y) format.<BR>\n" );
document.write( "you will find that the minimum cost will be at (25,50).\r<BR>\n" );
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document.write( "all the constraints must be met at that coordinate point as well.\r<BR>\n" );
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document.write( "the graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/122002.jpg\" >\r<BR>\n" );
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document.write( "at (x,y) = (25,50), the cost is 25 * 20000 + 50 * 25000 = 1,750,000.<BR>\n" );
document.write( "the number of days are 25 at refinery 1 and 50 at refinery 2.<BR>\n" );
document.write( "the number of barrels of high grade oil are 400 * 25 + 300 * 50 = 25,000<BR>\n" );
document.write( "the number of barrels of medium grade oil are 300 * 25 + 400 * 50 = 27,500<BR>\n" );
document.write( "the number of barrels of low grade oil are 200 * 25 + 500 * 50 = 30,000.\r<BR>\n" );
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document.write( "i should have graphed x <= 0 and y <= 0.<BR>\n" );
document.write( "in this case, it didn't matter, because those inequalities are already shaded.\r<BR>\n" );
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document.write( "let me know if you have any questions.\r<BR>\n" );
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document.write( "remember, with the desmos.com calculator, you graph the opposite of the inequalities.<BR>\n" );
document.write( "this is necessary so that the feasible region is not shaded.<BR>\n" );
document.write( "trying to graph the inequalities as they are will result in shading that overlaps other shading.<BR>\n" );
document.write( "that makes identifying the feasible region more difficult.\r<BR>\n" );
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document.write( "if you we to manually create the graph, then shading the inequalities as they are makes more sense, because you can control the shading better.<BR>\n" );
document.write( "it's also more work.<BR>\n" );
document.write( "doing it by using the desmos.com software is much easier.\r<BR>\n" );
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document.write( "theo<BR>\n" );
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