document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1123219>Question 1123219</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A steel company has two mills. Mill 1 costs $70,000 per day to operate, and it can produce 400 tons of high-grade steel, 500 tons of medium-grade steel, and 450 tons of low-grade steel each day. Mill 2 costs $60,000 per day to operate, and it can produce 350 tons of high-grade steel, 600 tons of medium-grade steel, and 400 tons of low-grade steel each day. The company has orders totaling 100,000 tons of high-grade steel, 150,000 tons of medium-grade steel, and 124,500 tons of low-grade steel. How many days should the company run each mill to minimize its costs and still fill the orders?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #739841 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=739841\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x = number of days to operate mill 1.<BR>\n" );
document.write( "y = number of days to operate mill 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your objective function is 70,000 * x + 60,000 * y.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you will evaluate this objective function at each of the corner points of the feasible region on the graph to find the corner point that has the least cost.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your constraint functions are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "400x + 350y >= 100,000 (hi grade steel requirement)<BR>\n" );
document.write( "500x + 600y >= 150,000 (medium grade steel requirement)<BR>\n" );
document.write( "450x + 400y >= 124,500 (lo grade steel requirement)<BR>\n" );
document.write( "x,y >= 0 (number of days can't be negative)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "using the desmos.com calculator, you will graph the opposite of these inequalities.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the area on the graph that is not shaded is your region of feasibility.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the corner points of this region are where the minimum cost will be found.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the graph looks like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/091704.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the corner points are (0,311.25), (210,75), (300,0)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the cost at (0,311.25) is 0 * 70,000 + 311.25 * 60,000 = 18,675,000.<BR>\n" );
document.write( "the cost at (310,75) is 210 * 70,000 + 75 * 60,000 = 19,200,000.<BR>\n" );
document.write( "the cost at (300,0) is 300 * 70,000 + 0 * 60,000 = 21,000,000.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the minimum cost is when you run mill2 for 311.25 daya.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "all the constraint functions need to be met when x = 0 and y = 311.25\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "400x + 350y = 108937.5 which is > 100,000.<BR>\n" );
document.write( "500x + 600y = 186,750 which is > 150,000.<BR>\n" );
document.write( "450x + 400y = 124,500 which is e3qual to 124,500.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "both x and y are > 0.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "all the constraint are met and the cost is minimum when x = 0 and y = 311.25.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that means mill 2 gets a lot of work and mill 1 doesn't get any.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );