document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1066667>Question 1066667</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A factory can produce two products, x and y, with a profit approximated by P =<BR>\n" );
document.write( "14x + 24y – 900. The production of y must exceed the production of x by at least 100 units. Moreover, production levels are limited by the formula x + 2y &#8804; 1400.\r<BR>\n" );
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document.write( "a. Identify the vertices of the feasible region.\r<BR>\n" );
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document.write( "b. What production levels yield the maximum profit, and what is the maximum 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 #681904 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=681904\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your profit equation is p = 14x + 24y - 900\r<BR>\n" );
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document.write( "your constraint equations are:\r<BR>\n" );
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document.write( "x + 2y <= 1400<BR>\n" );
document.write( "y >= x + 100<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "in the www.desmos.com calculator, you can do the following.\r<BR>\n" );
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document.write( "graph the REVERSE inequalities.\r<BR>\n" );
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document.write( "if the equation says > you would graph <.<BR>\n" );
document.write( "if the equation says >=, you would graph <=.<BR>\n" );
document.write( "if the equation says =, you would graph =.\r<BR>\n" );
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document.write( "the software will shade the areas of the graph that are NOT in the feasible region.\r<BR>\n" );
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document.write( "the region of feasible region will be the areas of the graph that are not shaded.\r<BR>\n" );
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document.write( "you could do it directly, but spotting the region of feasibility is easier when you graph the reverse inequalities.\r<BR>\n" );
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document.write( "try it yourself.\r<BR>\n" );
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document.write( "you will see what i mean.\r<BR>\n" );
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document.write( "in fact, i'll show you two graphs.\r<BR>\n" );
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document.write( "the first graphs the opposite of the region of feasibility.\r<BR>\n" );
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document.write( "the second graphs the region of feasibility.\r<BR>\n" );
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document.write( "in the first graph, you will graph:\r<BR>\n" );
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document.write( "x + 2y >= 1400<BR>\n" );
document.write( "y <= x + 100<BR>\n" );
document.write( "x <= 0<BR>\n" );
document.write( "y <= 0\r<BR>\n" );
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document.write( "this is opposite of the region of feasibility.<BR>\n" );
document.write( "the unshaded area is the region of feasibility.\r<BR>\n" );
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document.write( "in the second graph, you will graph:\r<BR>\n" );
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document.write( "x + 2y <= 1400<BR>\n" );
document.write( "y >= x + 100<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "the shaded areas is the region of feasibility.\r<BR>\n" );
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document.write( "desmos calculator can be found at www.desmos.com/calculator\r<BR>\n" );
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document.write( "in both cases, the vertices of the region of feasibility are:\r<BR>\n" );
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document.write( "(0,700)<BR>\n" );
document.write( "(0,100)<BR>\n" );
document.write( "(400,500)\r<BR>\n" );
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document.write( "you will analyze the objective function of P = 14x + 24y - 900 at each of the intersection points.\r<BR>\n" );
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document.write( "you will get:\r<BR>\n" );
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document.write( "(0,700) = 15,900<BR>\n" );
document.write( "(0,100) = 1,500<BR>\n" );
document.write( "(400,500) = 16,700\r<BR>\n" );
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document.write( "your maximum profit is when x = 400 and y = 500\r<BR>\n" );
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document.write( "the constraints also have to be satisfied.\r<BR>\n" );
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document.write( "y must be greater than x by 100.<BR>\n" );
document.write( "since y = 500 and x = 400, this constraint is satisfied.\r<BR>\n" );
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document.write( "x + 2y must be smaller than or equal to 1400.<BR>\n" );
document.write( "since  = 400 and y = 500, then x + 2y = 400 + 1000 = 1400.<BR>\n" );
document.write( "this constraint is satisfied also.\r<BR>\n" );
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document.write( "the graphs are shown below:<BR>\n" );
document.write( "first graph shows inequalities opposite to those required.<BR>\n" );
document.write( "second graph shows inequalities directly to those required.<BR>\n" );
document.write( "first graph is easier to see the region of feasibility in my opinion.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/020301.jpg\" alt=\"RRR\" </>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src= \"http://theo.x10hosting.com/2017/020302.jpg\" alt=\"RRR\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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