document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=250218>Question 250218</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A clothing manufacturer has 100 m of silk and 180 m of wool.<BR>\n" );
document.write( "To make a suit requires 2m of silk and 3m of wool, and to make a dress requires 1m of silk and 2m of wool. If the profit on a suit is $108 and the profit on a dress is $60, how many suites and dresses should the manufacturer make to maximize a profit?<BR>\n" );
document.write( "My answer <BR>\n" );
document.write( "Let P Rep the profit<BR>\n" );
document.write( "Let X Rep the number of Suits<BR>\n" );
document.write( "Let Y Rep the number of Dresses<BR>\n" );
document.write( "Constraints<BR>\n" );
document.write( "2x+1y < or = 100<BR>\n" );
document.write( "3x +2y < or = 180<BR>\n" );
document.write( "x< or = 0<BR>\n" );
document.write( "y< or = 0\r<BR>\n" );
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document.write( "p=108x + 60y\r<BR>\n" );
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document.write( "2x+1y=100   *3<BR>\n" );
document.write( "3x+2y=180   *2<BR>\n" );
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document.write( "6x+3y=300<BR>\n" );
document.write( "6x+4y=360<BR>\n" );
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document.write( "-1y= -60<BR>\n" );
document.write( "y=60\r<BR>\n" );
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document.write( "2x+1(60)=100<BR>\n" );
document.write( "2x=40<BR>\n" );
document.write( "x=20\r<BR>\n" );
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document.write( "the manufacturer should make 20 suits and 60 dresses to maximize profit  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #182160 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=182160\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "You got the correct answer, but I think you got there by chance.  If you graph the  four constraint inequalities, you will see that you have an area of feasibility described by the quadrilateral with vertices (0,0), (0,90), (20,60), and (50,0).\r<BR>\n" );
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document.write( "An elementary Linear Optimization theorem says that the optimum values are at a vertex of the feasibility polygon.  Because of the particular profit values given, the actual optimum for this problem occurs at the intersection of the two resource constraint boundary lines.  But if the profit numbers change, you could have a different result.  Completing this problem requires that you check the value of the Profit function for EACH of the feasibility vertices to verify that indeed you have chosen the one that optimizes profit.\r<BR>\n" );
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document.write( "As I mentioned, your answer is correct because:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 108(20)\ +\ 60(60)\ =\ 5760\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 108(0)\ +\ 60(90)\ =\ 5400\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 108(50)\ +\ 60(0)\ =\ 5400\">\r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 108(0)\ +\ 60(0)\ =\ 0\">\r<BR>\n" );
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document.write( "However, if the profit numbers change you can get a different result.  Let the profit on suits remain at 108, but the profit on dresses increase to 75.  Then:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 108(20)\ +\ 75(60)\ =\ 6660\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 108(0)\ +\ 75(90)\ =\ 6750\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 108(50)\ +\ 75(0)\ =\ 5400\">\r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 108(0)\ +\ 60(0)\ =\ 0\">\r<BR>\n" );
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document.write( "And you would want to manufacture zero suits and 90 dresses.\r<BR>\n" );
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document.write( "On the other hand, if the profit on suits were to increase to 130 leaving the profit on dresses at 60, then:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 130(20)\ +\ 60(60)\ =\ 6200\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 130(0)\ +\ 60(90)\ =\ 5400\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 130(50)\ +\ 60(0)\ =\ 6500\">\r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 108(0)\ +\ 60(0)\ =\ 0\">\r<BR>\n" );
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document.write( "So you would want to manufacture 50 suits and zero dresses.\r<BR>\n" );
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document.write( "Lesson:  Check all your possible solutions before making a judgement.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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