document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184212>Question 1184212</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Diana runs a factory that makes DVD players. Each R80 takes 6 ounces of plastic and 2 ounces of metal. Each FS20 requires 2 ounces of plastic and 4 ounces of metal. The factory has 140 ounces of plastic, 200 ounces of metal available, with a maximum of 12 R80 that can be built each week. If each R80 generates $5 in profit, and each FS20 generates $7, how many of each of the DVD players should Diana have the factory make each week to make the most profit?\r<BR>\n" );
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document.write( "R80: \r<BR>\n" );
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document.write( "FS20: \r<BR>\n" );
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document.write( "Best profit: <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 #815204 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=815204\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> make a table such as the one below:\r<BR>\n" );
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document.write( "product name           R80           FS20\r\n" );
document.write( "number of units         x              y        x >= 0, y >= 0, x <= 12\r\n" );
document.write( "plastic                 6              2        6x + 2y <= 140\r\n" );
document.write( "metal                   2              4        2x + 4y <= 200\r\n" );
document.write( "profit                  5              7        maximize 5x + 7y\r\n" );
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document.write( "</PRE>\r<BR>\n" );
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document.write( "objective function is profit = 5x + 7y<BR>\n" );
document.write( "this is what you want to maximize.\r<BR>\n" );
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document.write( "constraint functions:<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0<BR>\n" );
document.write( "x <= 12<BR>\n" );
document.write( "6x + 2y <= 140<BR>\n" );
document.write( "2x + 4y <= 200\r<BR>\n" );
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document.write( "using the <a  rel=nofollow HREF= \"https://www.desmos.com/calculator\" target = \"_blank\">desmos.com calculator</a>, you graph the opposite of the constraint functions and evaluate each corner point of the feasible region with the objective function.\r<BR>\n" );
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document.write( "the feasible region is the area of the graph that is not shaded.\r<BR>\n" );
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document.write( "here's the graph.\r<BR>\n" );
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document.write( "the corner points of the feasible region and the value of the objective function at those corner points are:\r<BR>\n" );
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document.write( "(0,50) = 5x + 7y becomes 5*0 + 7*50 = 350<BR>\n" );
document.write( "(8,46) = 5x + 7y becomes 5*8 + 46*7 = 362 ***** maximum profit is here.<BR>\n" );
document.write( "(12,34) = 5x + 7y becomes 5*12 + 7*34 = 298<BR>\n" );
document.write( "(12,0) = 5x + 7y becomes 5*12 + 7*0 = 60\r<BR>\n" );
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document.write( "the maximum profit is at (8,46), where it is equal to 362.<BR>\n" );
document.write( "(8,46) means the value of x is 8 and the value of y is 46.\r<BR>\n" );
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document.write( "the constraint functions must all be true at this point.\r<BR>\n" );
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document.write( "x <= 12 is true.<BR>\n" );
document.write( "x >= 0, y >= 0 is true.<BR>\n" );
document.write( "6x + 2y becomes 6*8 + 2*46 = 140 which is <= 140, so this is true.<BR>\n" );
document.write( "2x + 4y becomes 2*8 + 4*46 = 200 which is <= 200, so this is true.\r<BR>\n" );
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document.write( "all constraints are true at the maximum profit point of (8,46).<BR>\n" );
document.write( "this confirms the solution at (8,46) is valid.\r<BR>\n" );
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