document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181641>Question 1181641</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An oil refinery refines types A and B of crude oil and can refine as much as 4000 barrels each week. Type A crude has 2 kg of impurities per barrel, type B has 3 kg of impurities per barrel,and the refinery can handle no more than 9000 kg of these impurities each week.How much of each type should be refined in order to maximize profits, if the profit is R25/barrel for type A and R30/barrel for type B? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849998 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849998\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The AI solution from the other tutor is correct.<br><BR>\n" );
document.write( "In that solution, the steps outlined say that you need to evaluate the objective function at each corner of the feasibility region.  And that is what most references say.<br><BR>\n" );
document.write( "However, in fact it is NOT necessary to evaluate the objective function at every corner of the feasibility region.<br><BR>\n" );
document.write( "The corner of the feasibility region where the objective function is maximized can be determined from the slopes of the constraint boundary lines and of the objective function.  The idea there is that the objective function will be maximized where a line with the slope of the objective function just touches the feasibility region.<br><BR>\n" );
document.write( "In this problem, the constraint boundary line equations are x+y=4000 (slope -1) and 2x+3y=9000 (slope -2/3); the objective function is P=25x+30y (slope -5/6).  Since -5/6 is between -1 and -2/3, the objective function is maximized at the intersection of the two constraint boundary lines.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=4000 ALIGN=MIDDLE  ALT=\"x%2By=4000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B3y=9000 ALIGN=MIDDLE  ALT=\"2x%2B3y=9000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B2y=8000 ALIGN=MIDDLE  ALT=\"2x%2B2y=8000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1000 ALIGN=MIDDLE  ALT=\"y=1000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3000 ALIGN=MIDDLE  ALT=\"x=3000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: Profit is maximized when 3000 barrels of type A and 1000 barrels of type B are refined each week.<br><BR>\n" );
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