document.write( "Question 1131043: You have exactly 36 units of red dye and 28 units of blue dye. Your boss also wants you to make as many gallons of unique color \"c\" and unique color \"d\" as possible. Each gallon of color c requires 1 unit of red dye and 2 units of blue dye. Each gallon of color d requires 6 units of red dye and 1 unit of blue dye. Find the most amount of gallons you can create of both color c and color d \n" ); document.write( "

Algebra.Com's Answer #747663 by ikleyn(52817) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "I will solve the problem using the Linear Programming method.\r
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document.write( "Let X and Y be the numbers of gallons of liquid C and D, respectively.\r\n" );
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document.write( "Then the objective function is  F(X,Y) = X + Y.\r\n" );
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document.write( "The constraint inequalities are\r\n" );
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document.write( "1*x + 6*Y <= 36     (1)  (red dye constraint)\r\n" );
document.write( "2*X + 1*Y <= 28     (2)  (blue dye constraint)\r\n" );
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document.write( "X >= 0,  Y >= 0.\r\n" );
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document.write( "The feasibility region is shown in the figure below.\r\n" );
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document.write( "It is the quadrilateral in QI, adjacent to x- and y- axes and constrained by the red and the green lines.\r\n" );
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document.write( "Plots 1X + 6Y = 36 (red)  and  2X + Y = 28 (green)\r\n" );
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document.write( "The feasibility quadrilateral has the corner  \r\n" );
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document.write( "    P1 = (0,6)      (red line y-intercept);\r\n" );
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document.write( "    P2 = (12,4)     (intersection point of the red and green lines);  and\r\n" );
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document.write( "    P3 = (14,0)     (green line x-intercept).\r\n" );
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document.write( "The objective function has the values\r\n" );
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document.write( "     at P1:  F(X,Y) = 0 + 6 = 6;\r\n" );
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document.write( "     at P2:  F(X,Y) = 12 + 4 = 16;   and\r\n" );
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document.write( "     at P3:  F(X,y) = 14 + 0 = 14.\r\n" );
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document.write( "According to the Linear programming method, it means that the point P2 gives the solution to the given linear minimax problem:\r\n" );
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document.write( "    The most amount of gallons you can create of both color C and color D is 16.\r\n" );
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document.write( "    It happens when 12 gallons of the liquid C and 4 gallons of the liquid D is produced.\r\n" );
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\n" ); document.write( "\n" ); document.write( "If you want to see other similar minimax problems, solved by the Linear programming method, look into the lesson\r
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