document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198872>Question 1198872</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manufacturer produces two models of mountain bikes. Model A requires 5 hours of assembly time and 2 hours of painting time, and Model B requires 4 hours of assembly time and 3 hours of painting time. The maximum total weekly hours available in the assembly department and the painting department are 200 hours and 108 hours, respectively. The profits per unit are $25 for Model A and $15 for Model B. How many of each type should be produced to maximize profit?\r<BR>\n" );
document.write( "\n" );
document.write( "Bikes of Model A=\r<BR>\n" );
document.write( "\n" );
document.write( "Bikes of Model B= </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832564 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=832564\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x = number of model A bikes.<BR>\n" );
document.write( "let y = number of model B bikes.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "objective function is:<BR>\n" );
document.write( "profit = 25x + 15y\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "constraint inequalities are:<BR>\n" );
document.write( "5x + 4y <= 200<BR>\n" );
document.write( "2x + 3y <= 108\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "it helps to make a chart such as the one below:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( "                       model A           model B\r\n" );
document.write( "number of bikes           x                 y\r\n" );
document.write( "assembly time             5                 4            <= 200\r\n" );
document.write( "painting time             2                 3            <= 108\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this way it's easier to see what hours belong where.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "additional constraints are that x and y must both be greater than or equal to 0.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "using the desmos.com calculator, you would graph the opposites of the inequalities.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the area of the graph that is not shaded would be the region of feasibility.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you would evaluate the objective function at the corner points of this region.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the maximum profit will be at one of those corner points.   \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "after evaluating the objective function at each corner pooint, you will find that the maximum profit is at (40,0).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this means 40 model A mountain bikes and no model B mountain bikes should be made to maximize profit, assuming all mountain bikes are sold.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the constraint inequalities must also be satisfied at each corner point.<BR>\n" );
document.write( "at (40,0), .....\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "assembly time would be equal to 40 * 5 = 200<BR>\n" );
document.write( "painting time would be equal to 40 * 2 = 80\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "both of these are within limits.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this is what the graph looks like.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2022/120403.jpg\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the desmos.com graphing software can be found at <a  rel=nofollow HREF= \"https://www.desmos.com/calculator\" target = \"_blank\">https://www.desmos.com/calculator</a> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );