document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1041919>Question 1041919</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manufacturing company receives orders for engines from two assembly plants. Plant I needs at least 45 &#8203;engines, and plant II needs at least 32 engines. The company can send at most 120 engines to these assembly plants. It costs &#8203;$30 per engine to ship to plant I and &#8203;$40 per engine to ship to plant II. Plant I gives the manufacturing company &#8203;$20 in rebates toward its products for each engine they&#8203; buy, while plant II gives similar &#8203;$10 rebates. The manufacturer estimates that they need at least &#8203;$1500 in rebates to cover products they plan to buy from the two plants. How many engines should be shipped to each plant to minimize shipping&#8203; costs? What is the minimum&#8203; cost?<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 #659195 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=659195\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your objective function is to minimize the cost.\r<BR>\n" );
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document.write( "set x equal to the number of engines shipped to plant I.<BR>\n" );
document.write( "set y equal to the number of engines shipped to plant II.\r<BR>\n" );
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document.write( "total cost = 30x + 40y\r<BR>\n" );
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document.write( "that's the function you want to minimize.<BR>\n" );
document.write( "it costs 30 dollars to ship 1 engine to plant I.<BR>\n" );
document.write( "it costs 40 dollars to ship 1 engine to plant II.\r<BR>\n" );
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document.write( "set c = total cost.\r<BR>\n" );
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document.write( "your objective function is c = 30x + 40y.\r<BR>\n" );
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document.write( "your constraints are:\r<BR>\n" );
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document.write( "x >= 45<BR>\n" );
document.write( "y >= 32\r<BR>\n" );
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document.write( "this means you need to ship at least 45 engines to plant I and at least 32 engines to plant II.\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( "this means that the number of engines shipped has to be equal to or greater than 0.\r<BR>\n" );
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document.write( "20x + 10y >= 1500\r<BR>\n" );
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document.write( "this means that the total rebates provided by the assembly plants has to be greater than or equal to 1500.\r<BR>\n" );
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document.write( "x + y < = 120\r<BR>\n" );
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document.write( "this means that the total number of engines that can be shipped has to be less than or equal to 120.\r<BR>\n" );
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document.write( "summarize your formulas:\r<BR>\n" );
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document.write( "objective function:\r<BR>\n" );
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document.write( "c = 30x + 40y\r<BR>\n" );
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document.write( "constraints:\r<BR>\n" );
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document.write( "x + y <= 120<BR>\n" );
document.write( "x >= 45<BR>\n" );
document.write( "y >= 32<BR>\n" );
document.write( "20x + 10y >= 1500\r<BR>\n" );
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document.write( "graph the constraint equations and then shade the areas on the graph that satisfy the inequalities.\r<BR>\n" );
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document.write( "this is what my graph looks like.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/081401.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "i graphed the equations.<BR>\n" );
document.write( "i then shaded the areas that satisfied the inequalities.\r<BR>\n" );
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document.write( "then you need to find the corner points of the shaded area.\r<BR>\n" );
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document.write( "those are:\r<BR>\n" );
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document.write( "(45,75)<BR>\n" );
document.write( "(45,60)<BR>\n" );
document.write( "(59,32)<BR>\n" );
document.write( "(88,32)\r<BR>\n" );
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document.write( "you then analyze the objective function at those corner points.\r<BR>\n" );
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document.write( "objective function is c = 30x + 40y\r<BR>\n" );
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document.write( "at (45,75), you get 30*45 + 40*75 = 4350\r<BR>\n" );
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document.write( "at (45,60), you get 30*45 + 40*60 = 3750\r<BR>\n" );
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document.write( "at (59,32), you get 30*59 + 40*32 = 3050\r<BR>\n" );
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document.write( "at (88,32), you get 30*88 + 40*32 = 3920\r<BR>\n" );
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document.write( "your minimum cost is when x = 59 and y = 32.\r<BR>\n" );
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document.write( "your minimum cost is 3050.\r<BR>\n" );
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document.write( "all your constraints need to be satisfied as well.\r<BR>\n" );
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document.write( "x + y <= 120 becomes 59 + 32 <= 120 which becomes 91 <= 120 which is true.\r<BR>\n" );
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document.write( "x >= 45 becomes 59 >= 45 which is true.\r<BR>\n" );
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document.write( "y >= 32 becomes 32 >= 32 which is true.\r<BR>\n" );
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document.write( "x and y >= 0 are automatically satisfied when x >= 45 and y >= 32.\r<BR>\n" );
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document.write( "20x + 10y >= 1500 becomes 59*20 + 32*10 >= 1500 which becomes 1180 + 320 >= 1500 which becomes 1500 >= 1500 which is true.\r<BR>\n" );
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document.write( "all the constraints are satisfied.\r<BR>\n" );
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document.write( "your solution is that the minimum cost is 3050.\r<BR>\n" );
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