document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172680>Question 1172680</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A Papyrus company owns two automated machines. Machine A costs ₱25,000 per day<BR>\n" );
document.write( "to operate, and it can produce 40 rims of high-grade paper, 30 rims of medium-grade<BR>\n" );
document.write( "paper, and 20 rims of low-grade paper each day. Machine B costs ₱35,000 per day to<BR>\n" );
document.write( "operate, and it can produce 30 rims of high-grade paper, 40 rims of medium-grade<BR>\n" );
document.write( "paper, and 50 rims of low-grade paper each day. The company has orders totaling<BR>\n" );
document.write( "2,000 rims of high-grade paper, 2,000 rims of medium-grade paper, and 3,000 rims of<BR>\n" );
document.write( "low-grade paper. How many days should it run each machine to minimize its costs but<BR>\n" );
document.write( "have enough supply of paper for 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 #797789 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=797789\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the graphical solution is shown below.\r<BR>\n" );
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document.write( "the simplex method tool solution is shown below:\r<BR>\n" );
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document.write( "both tell you that the optimum solution is when x = 7.143 and y = 57.143.<BR>\n" );
document.write( "the graphical solution is rounded to 3 decimal places.<BR>\n" );
document.write( "the simplex method tool solution rounds to 6 significant digits.<BR>\n" );
document.write( "rounding to decimal digits only includes the fractional part of the answer.<BR>\n" );
document.write( "rounding to significant digits includes the whole and fractional part of the answer.\r<BR>\n" );
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document.write( "the minimal cost is 2178580 from the graphical solution and 2178570 from the simplex method tool solution.\r<BR>\n" );
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document.write( "with the graphical solution, using the desmos.com calculator, you graph the opposite of the constraint inequalities.<BR>\n" );
document.write( "the region of feasibility is the area on the graph that is not shaded.<BR>\n" );
document.write( "the corner points of the region of feasibility are where the optimum solutions lie.<BR>\n" );
document.write( "you evaluate the objective function at each of these corner points to find the minimal cost solution.\r<BR>\n" );
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document.write( "the graphical solution shows all the possible values of x and y, one of which will provide the optimal solution.<BR>\n" );
document.write( "the simplex method tool only shows optimal solution.\r<BR>\n" );
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document.write( "all the constraints need to be satisfied at the optimal solution.<BR>\n" );
document.write( "with (7.143, 57.143):<BR>\n" );
document.write( "40x + 30y = 2000.01 which is >= 2000.<BR>\n" );
document.write( "30x + 40y = 2500.01 which is >= 2000.<BR>\n" );
document.write( "20x + 50y = 3000.01 which is >= 3000.<BR>\n" );
document.write( "keep in mind that these are rounded numbers, so the answers won't necessarily be right on.\r<BR>\n" );
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document.write( "with the simplex method tool, you can set the rounding.<BR>\n" );
document.write( "for example, it was set at 6 significant digits.<BR>\n" );
document.write( "this will get you closer to the actual results.<BR>\n" );
document.write( "it can also be set to greater than 6 significant digits.<BR>\n" );
document.write( "i set it to 12 significant digits and this is what is said:<BR>\n" );
document.write( "Optimal Solution: c = 2178571.42857; x = 7.14285714286, y = 57.1428571429\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>\r<BR>\n" );
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document.write( "the simplex method tool can be found at <a  rel=nofollow HREF= \"https://www.zweigmedia.com/RealWorld/simplex.html\" target = \"_blank\">https://www.zweigmedia.com/RealWorld/simplex.html</a>\r<BR>\n" );
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document.write( "your objective function was:<BR>\n" );
document.write( "cost = 25000x + 35000y<BR>\n" );
document.write( "this you want to minimize.\r<BR>\n" );
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document.write( "your constraint inequalities were:<BR>\n" );
document.write( "40x + 30y >= 2000 (hi)<BR>\n" );
document.write( "30x + 40y >= 2000 (med)<BR>\n" );
document.write( "20x + 50y >= 3000 (lo)<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "i'll be available to answer any questions you might have regardng this.<BR>\n" );
document.write( "theo\r<BR>\n" );
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