document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186205>Question 1186205</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The Intellectual Company produces a chemical solution used for cleaning carpets. This chemical is made from a mixture of two other chemicals which contain cleaning agent X and cleaning agent Y. Their product must contain 175 units of agent X and 150 units of agent Y and weigh at least 100 pounds. Chemical A costs ₱ 8 per pound, while chemical B costs ₱ 6 per pound. Chemical A contains one unit of agent X and three units of agent Y. Chemical B contains seven units of agent X and one unit of agent Y.\r<BR>\n" );
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document.write( "a. Set up the following: <BR>\n" );
document.write( "i. Variables<BR>\n" );
document.write( "ii. Constraints<BR>\n" );
document.write( "iii. Objective Function<BR>\n" );
document.write( "b. Find the minimum cost <BR>\n" );
document.write( "c. Determine the best combination of the ingredients to minimize the 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 #849470 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=849470\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to set up and solve this linear programming problem:\r<BR>\n" );
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document.write( "**a. Set up:**\r<BR>\n" );
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document.write( "**i. Variables:**\r<BR>\n" );
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document.write( "*   x = pounds of chemical A<BR>\n" );
document.write( "*   y = pounds of chemical B\r<BR>\n" );
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document.write( "**ii. Constraints:**\r<BR>\n" );
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document.write( "*   Agent X:  x + 7y ≥ 175<BR>\n" );
document.write( "*   Agent Y:  3x + y ≥ 150<BR>\n" );
document.write( "*   Weight: x + y ≥ 100<BR>\n" );
document.write( "*   Non-negativity: x ≥ 0, y ≥ 0\r<BR>\n" );
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document.write( "**iii. Objective Function:**\r<BR>\n" );
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document.write( "Minimize Cost (C) = 8x + 6y\r<BR>\n" );
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document.write( "**b. Find the minimum cost:**\r<BR>\n" );
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document.write( "1.  **Graph the constraints:** Treat each inequality as an equation and plot the lines. Shade the appropriate region based on the inequality.  For example, for x + 7y ≥ 175, plot the line x + 7y = 175, and shade the region *above* and to the *right* of the line. Do this for all constraints.\r<BR>\n" );
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document.write( "2.  **Identify the feasible region:** The feasible region is the area where all the shaded regions overlap.\r<BR>\n" );
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document.write( "3.  **Find the vertices:** The vertices of the feasible region are the points where the constraint lines intersect. Solve systems of equations to find these intersection points.  The relevant vertices are:\r<BR>\n" );
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document.write( "    *   Intersection of x + 7y = 175 and 3x + y = 150: Solving these gives x = 43, y = 19<BR>\n" );
document.write( "    *   Intersection of 3x + y = 150 and x + y = 100: Solving these gives x = 25, y = 75<BR>\n" );
document.write( "    *   Intersection of x + 7y = 175 and x + y = 100: Solving these gives x = 12.5, y = 87.5\r<BR>\n" );
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document.write( "4.  **Evaluate the objective function at each vertex:**\r<BR>\n" );
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document.write( "    *   C(43, 19) = 8(43) + 6(19) = 344 + 114 = ₱458<BR>\n" );
document.write( "    *   C(25, 75) = 8(25) + 6(75) = 200 + 450 = ₱650<BR>\n" );
document.write( "    *   C(12.5, 87.5) = 8(12.5) + 6(87.5) = 100 + 525 = ₱625\r<BR>\n" );
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document.write( "5.  **Determine the minimum cost:** The minimum cost is the smallest value of the objective function.\r<BR>\n" );
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document.write( "The minimum cost is ₱458.\r<BR>\n" );
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document.write( "**c. Determine the best combination of ingredients:**\r<BR>\n" );
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document.write( "The best combination of ingredients is the (x, y) values that correspond to the minimum cost.\r<BR>\n" );
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document.write( "The minimum cost of ₱458 occurs when x = 43 and y = 19.\r<BR>\n" );
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document.write( "Therefore, the best combination is **43 pounds of chemical A and 19 pounds of chemical B**.<BR>\n" );
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