document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1142002>Question 1142002</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A company manufactures x units of Product A and y units of Product B, on two machines, I and II. It has been determined that the company will realize a profit of $6/unit of Product A and a profit of $7/unit of Product B. To manufacture a unit of Product A requires 6 min on Machine I and 5 min on Machine II. To manufacture a unit of Product B requires 9 min on Machine I and 4 min on Machine II. There are 5 hr of machine time available on Machine I and 3 hr of machine time available on Machine II in each work shift. How many units of each product should be produced in each shift to maximize the company's profit?<BR>\n" );
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document.write( "What is the optimal profit? (Round your answer to the nearest whole number.)<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 #762669 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=762669\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your objective function is 6x + 7y = profit\r<BR>\n" );
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document.write( "your constraint functions are:\r<BR>\n" );
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document.write( "6x + 9y <= 300<BR>\n" );
document.write( "5x + 4y <= 180\r<BR>\n" );
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document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "all constraints have to be in minutes.\r<BR>\n" );
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document.write( "5 hours = 300 minutes<BR>\n" );
document.write( "3 hours = 180 minutes\r<BR>\n" );
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document.write( "it helps to set up a table like the one below that allows you to visualize what's happening.\r<BR>\n" );
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document.write( "make sure that all constraints are dealing with the same units.<BR>\n" );
document.write( "constraint units are in minutes for this problem.\r<BR>\n" );
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document.write( "                                product A      product B\r\n" );
document.write( "number of units                     x              y  \r\n" );
document.write( "profit                              6              7          maximize\r\n" );
document.write( "machine 1                           6              9          <= 300\r\n" );
document.write( "machine 2                           5              4          <= 180\r\n" );
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document.write( "your maximum profit will be when 20 units of product A and 20 units of product B are produced.\r<BR>\n" );
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document.write( "the maximum profit is 260 dollars.\r<BR>\n" );
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document.write( "you can solve this by using a linear optimization program such as 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>, or by using the solver that comes with excel (it's an add in on the version of excel that i'm using), or by graphing.\r<BR>\n" );
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document.write( "i did all three and got the same answer, so it looks to be good, assuming i set the problem up correctly.\r<BR>\n" );
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document.write( "the linear optimization simplex solution is shown below.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/060711.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the excel solution is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/060712.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the graphical solution is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/060713.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the desmos.com graphing software makes solving these types of problems easy.\r<BR>\n" );
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document.write( "using this software, you would graph the opposite of the inequalities.\r<BR>\n" );
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document.write( "the area of the graph that is not shaded is the region of feasibility.\r<BR>\n" );
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document.write( "the corner points of the region of feasibility are where the value of the objective function are evaluated.\r<BR>\n" );
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document.write( "for example, at the coordinate point of (20,20), the objective function of 6x + 7y is evaluated to get 6*20 + 7*20 = 260.\r<BR>\n" );
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document.write( "all the constraint functions have to be satisfied as well.\r<BR>\n" );
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document.write( "x and y are greater than or equal to 0.\r<BR>\n" );
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document.write( "6x + 9y = 6*20 + 9*20 = 300 which is less than or equal to 300.\r<BR>\n" );
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document.write( "5x + 4y = 5*20 + 4*20 = 180 which is less than or equal to 180.\r<BR>\n" );
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