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 #849984 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849984\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to determine the optimal amount of each type of crude oil to refine:\r<BR>\n" );
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document.write( "**1. Define the variables:**\r<BR>\n" );
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document.write( "*   Let 'x' be the number of barrels of type A crude oil.<BR>\n" );
document.write( "*   Let 'y' be the number of barrels of type B crude oil.\r<BR>\n" );
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document.write( "**2. Set up the constraints:**\r<BR>\n" );
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document.write( "*   **Total barrels constraint:** x + y ≤ 4000  (The refinery can refine at most 4000 barrels per week)<BR>\n" );
document.write( "*   **Impurities constraint:** 2x + 3y ≤ 9000 (The refinery can handle no more than 9000 kg of impurities)<BR>\n" );
document.write( "*   **Non-negativity constraints:** x ≥ 0 and y ≥ 0 (We can't refine a negative number of barrels)\r<BR>\n" );
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document.write( "**3. Define the objective function (profit):**\r<BR>\n" );
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document.write( "Profit (P) = 25x + 30y  (We want to maximize this)\r<BR>\n" );
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document.write( "**4. Solve the linear programming problem:**\r<BR>\n" );
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document.write( "This can be solved graphically or using linear programming techniques (like the simplex method or software tools). Here's the general approach:\r<BR>\n" );
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document.write( "*   **Graphically:**<BR>\n" );
document.write( "    1.  Plot the constraints on a graph.<BR>\n" );
document.write( "    2.  Identify the feasible region (the area that satisfies all constraints).<BR>\n" );
document.write( "    3.  Find the corner points of the feasible region.<BR>\n" );
document.write( "    4.  Evaluate the objective function at each corner point.<BR>\n" );
document.write( "    5.  The corner point that yields the highest profit is the optimal solution.\r<BR>\n" );
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document.write( "*   **Linear Programming:** Specialized software or online calculators can solve this type of problem efficiently.\r<BR>\n" );
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document.write( "**Solution:**\r<BR>\n" );
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document.write( "The optimal solution is to refine:\r<BR>\n" );
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document.write( "*   3000 barrels of type A crude oil<BR>\n" );
document.write( "*   1000 barrels of type B crude oil\r<BR>\n" );
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document.write( "This will result in a maximum profit of R105,000.<BR>\n" );
document.write( " </SPAN>\n" );
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