document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1118663>Question 1118663</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Each day , a factory makes three types of widget : basic , standard and luxury. The widget produced need three different components : type A , type B and type C.<BR>\n" );
document.write( "Basic widget need 6 components of type A , 6 components type B and 12 components of type C.<BR>\n" );
document.write( "Standard widgets need 4 components of type A ,3 compliance of type B , and 18 components of type C.<BR>\n" );
document.write( "Luxury widgets need 2 components of type A , 9 components of type B and 6 components of type C. <BR>\n" );
document.write( "Each day , there are 240 components of type A available , 300 of type B. 900 of type C. <BR>\n" );
document.write( "Each day ,the factory must use at least twice as many components of type C as type B.<BR>\n" );
document.write( "Each day , the factory makes X basic widgets , y standard widgets and z luxury widgets. <BR>\n" );
document.write( "(a) In addition to x>\= 0 , y>\= 0 and z>\=0 , find four inequalities in x, y and z that model the above constraints , simplify each inequality. <BR>\n" );
document.write( "(b) Each day ,the factory makes the maximum possible number of widget . On a particular day , the factory must make the same number of luxury widgets as basic widgets. <BR>\n" );
document.write( "(i) show that your answers in part (a) become \r<BR>\n" );
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document.write( "      2x + y <\= 60<BR>\n" );
document.write( "      5x + y <\= 100 <BR>\n" );
document.write( "      X + y <\= 50 <BR>\n" );
document.write( "      Y >\= x  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #734044 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=734044\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you would create a table as follows:\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
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document.write( "type widget                       basic         standard         luxury\r\n" );
document.write( "number of widgets                   x               y               z\r\n" );
document.write( "number of type A components         6               4               2\r\n" );
document.write( "number of type B components         6               3               9\r\n" );
document.write( "number of type C components         12              18              6\r\n" );
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document.write( "the number of type A components has to be less than or equal to 240.\r<BR>\n" );
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document.write( "the equation for that is:\r<BR>\n" );
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document.write( "6x + 4y + 2z <= 240\r<BR>\n" );
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document.write( "this can be simplified to:\r<BR>\n" );
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document.write( "3x + 2y + z <= 120\r<BR>\n" );
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document.write( "the nunmber of type B components has to be less than or equal to 300.\r<BR>\n" );
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document.write( "the equation for that is:\r<BR>\n" );
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document.write( "6x + 3y + 9z <= 300\r<BR>\n" );
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document.write( "this can be simplified to:\r<BR>\n" );
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document.write( "2x + y + 3z <= 100\r<BR>\n" );
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document.write( "the number of type C components has to be less than or equal to 900.\r<BR>\n" );
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document.write( "the equation for that is:\r<BR>\n" );
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document.write( "12x + 18y + 6z <= 900\r<BR>\n" );
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document.write( "this can be simplified to:\r<BR>\n" );
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document.write( "2x + 3y + z <= 150\r<BR>\n" );
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document.write( "the number of type C components has to be greater than or equal to 2 times the nmber of type B components.\r<BR>\n" );
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document.write( "the equation for that is:\r<BR>\n" );
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document.write( "12x + 18y + 6z >= 2 * (6x + 3y + 9z)\r<BR>\n" );
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document.write( "this can be simplified to:\r<BR>\n" );
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document.write( "12x + 18y + 6z >= 12x + 6y + 18z\r<BR>\n" );
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document.write( "this can be simplified to:\r<BR>\n" );
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document.write( "2x + 3y + z >= 2x + y + 3z\r<BR>\n" );
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document.write( "your simplified equations are:\r<BR>\n" );
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document.write( "3x + 2y + z <= 120<BR>\n" );
document.write( "2x + y + 3z <= 100<BR>\n" );
document.write( "2x + 3y + z <= 150<BR>\n" );
document.write( "2x + 3y + z >= 2x + y + 3z\r<BR>\n" );
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document.write( "in part b, you are told that the factory must make the same number of luxury widgets as basic widgets.\r<BR>\n" );
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document.write( "this means that x must be equal to z.<BR>\n" );
document.write( "since x must be equal to z, you can replace z with x in the equations.<BR>\n" );
document.write( "you will get:\r<BR>\n" );
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document.write( "3x + 2y + x <= 120<BR>\n" );
document.write( "2x + y + 3x <= 100<BR>\n" );
document.write( "2x + 3y + x <= 150<BR>\n" );
document.write( "2x + 3y + x >= 2x + y + 3x\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "4x + 2y <= 120<BR>\n" );
document.write( "5x + y <= 100<BR>\n" );
document.write( "3x + 3y <= 150<BR>\n" );
document.write( "3x + 3y >= 5x + y\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "2x + y <= 60<BR>\n" );
document.write( "5x + y <= 100<BR>\n" );
document.write( "x + y <= 50\r<BR>\n" );
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document.write( "in 3x + 3y >= 5x + y, subtract 3x from both sides of the equation and subtract y from both sides of the equation to get:\r<BR>\n" );
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document.write( "3y - y >= 5x - 3x\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "2y >= 2x\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "y >= x\r<BR>\n" );
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document.write( "there you have it.\r<BR>\n" );
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document.write( "the 4 equations (they are really inequalities even though i called them equations) become:\r<BR>\n" );
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document.write( "2x + y <= 60<BR>\n" );
document.write( "5x + y <= 100<BR>\n" );
document.write( "x + y <= 50<BR>\n" );
document.write( "y >= x\r<BR>\n" );
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document.write( "that matches what you showed above, so this solution appears to be correct.\r<BR>\n" );
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document.write( "i used excel to come up with a solution, just to see what it would be.\r<BR>\n" );
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document.write( "the solution that excel came up with was 13 basic, 34 standard, 13 luxury.\r<BR>\n" );
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document.write( "that makes the maximum number of widgets to be produced equal to 60.\r<BR>\n" );
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document.write( "all the constraints were satisfied.\r<BR>\n" );
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document.write( "i forced all the number of widgets to be integers.\r<BR>\n" );
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document.write( "excel claims the answer, in that case, would be within 1% of the optimal solution.\r<BR>\n" );
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document.write( "the total number of widgets was the same when i didn't force the number of widgets to be integers.\r<BR>\n" );
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document.write( "the non-integer solution was x and z = 13.33333 and y = 33.33333<BR>\n" );
document.write( " </SPAN>\n" );
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