document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1123218>Question 1123218</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A small petroleum company owns two refineries. Refinery 1 costs $20,000 per day to operate, and it can produce 400 barrels of high-grade oil, 300 barrels of medium-grade oil, and 200 barrels of low-grade oil each day. Refinery 2 is newer and more modern. It costs $25,000 per day to operate, and it can produce 300 barrels of high-grade oil, 400 barrels of medium-grade oil, and 500 barrels of low-grade oil each day. The company has orders totaling 25,000 barrels of high-grade oil, 27,000 barrels of medium-grade oil, and 30,000 barrels of low-grade oil. How many days should it run each refinery to minimize its costs and still refine 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 #739838 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=739838\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x = the number of days to run refinery 1<BR>\n" );
document.write( "let y = the number of days to run refinery 2\r<BR>\n" );
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document.write( "your objective function is 20,000 * x + 25,000 * y which equals the total cost to provide the necessary amount of oil.\r<BR>\n" );
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document.write( "your constraint functions are:\r<BR>\n" );
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document.write( "400x + 300y >= 25000 (hi grade oil requirements)<BR>\n" );
document.write( "300x + 400y >= 27000 (medium grade oil requirements)<BR>\n" );
document.write( "200x +500y >= 30000 (lo grade oil requirements)<BR>\n" );
document.write( "x,y >= 0 (number of days can't be negative).\r<BR>\n" );
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document.write( "using the desmos.com calculator, you would graph the opposite of these inequalities.\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( "the minimum cost will be found by evaluating the objective function at each corner point of the region of feasibility.\r<BR>\n" );
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document.write( "the graph looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/091703.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the corner points are at (0,83.3333),(25,50),(150,0)\r<BR>\n" );
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document.write( "at (0,83.3333), the cost is 0 * 20,000 + 83.33 * 25,000 = 2,083,250.<BR>\n" );
document.write( "at (25,50), the cost is 25 * 20,000 + 50 * 25,000 = 1,750,000.<BR>\n" );
document.write( "at (150,0), the cost is 150 * 20,000 + 0 * 25,000 = 3,000,000\r<BR>\n" );
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document.write( "the minimum cost is achieved when refinery 1 is operated for 25 days and refinery 2 is operated for 50 days at a cost of 1,750,000.\r<BR>\n" );
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document.write( "the desmos.com calculator can be found at <a  rel=nofollow HREF= \"https://www.desmos.com/calculator\" target = \"_blank\">https://www.desmos.com/calculator</a> </SPAN>\n" );
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