document.write( "Question 1186248: 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
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\n" ); document.write( "\n" ); document.write( "b. Find the minimum cost
\n" ); document.write( "c. Determine the best combination of the ingredients to minimize the cost. \n" ); document.write( "
Algebra.Com's Answer #849463 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "The Intellectual Company produces a chemical solution used for cleaning carpets.
\n" ); document.write( "This chemical is made from a mixture of two other chemicals which contain cleaning agent X and cleaning agent Y.
\n" ); document.write( "Their product must contain 175 units of agent X and 150 units of agent Y and weigh at least 100 pounds.
\n" ); document.write( "Chemical A costs ₱ 8 per pound, while chemical B costs ₱ 6 per pound.
\n" ); document.write( "Chemical A contains one unit of agent X and three units of agent Y.
\n" ); document.write( "Chemical B contains seven units of agent X and one unit of agent Y.
\n" ); document.write( "a. Set up the following:
\n" ); document.write( "i. Variables
\n" ); document.write( "ii. Constraints
\n" ); document.write( "iii. Objective Function
\n" ); document.write( "b. Find the minimum cost
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\n" ); document.write( "\n" ); document.write( "        The solution in the post by  @CPhill,  giving the answer \r
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\n" ); document.write( "\n" ); document.write( "                x = 43 pounds of chemical A and y = 19 pounds of chemical B\r
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\n" ); document.write( "\n" ); document.write( "        is  INCORRECT.   It can be easily disproved,   since   x + y = 43 + 19 = 62,   which is less than  100.\r
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\n" ); document.write( "\n" ); document.write( "        Thus the restriction x+y >= 100 of the problem is not satisfied:  it is  FAILED,  instead.\r
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\n" ); document.write( "\n" ); document.write( "        The cause is that  @CPhill incorrectly determined the feasibility domain and used  WRONG  vertices for estimations.\r
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\n" ); document.write( "\n" ); document.write( "                   Below is my solution,  proper and correct.\r
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document.write( "**a. Set up:**\r\n" );
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document.write( "**i. Variables:**\r\n" );
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document.write( "*   x = pounds of chemical A\r\n" );
document.write( "*   y = pounds of chemical B\r\n" );
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document.write( "**ii. Constraints:**\r\n" );
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document.write( "*   Agent X:  x + 7y ≥ 175\r\n" );
document.write( "*   Agent Y:  3x + y ≥ 150\r\n" );
document.write( "*   Weight: x + y ≥ 100\r\n" );
document.write( "*   Non-negativity: x ≥ 0, y ≥ 0\r\n" );
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document.write( "**iii. Objective Function:**\r\n" );
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document.write( "Minimize Cost (C) = 8x + 6y\r\n" );
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document.write( "**b. Find the minimum cost:**\r\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.  \r\n" );
document.write( "      For example, for x + 7y ≥ 175, plot the line x + 7y = 175, and shade the region *above* and to the *right* of the lines. \r\n" );
document.write( "      Do this for all constraints.\r\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\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. \r\n" );
document.write( "      Solve systems of equations to find these intersection points.  The relevant vertices are:\r\n" );
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document.write( "    *   Intersection of x + 7y = 175 and     y =   0:                      x = 175,   y =  0\r\n" );
document.write( "    *   Intersection of x + 7y = 175 and x + y = 100:  Solving these gives x =  87.5, y = 12.5  \r\n" );
document.write( "    *   Intersection of 3x + y = 150 and x + y = 100:  Solving these gives x =  25,   y = 75\r\n" );
document.write( "    *   Intersection of 3x + y = 175 and x     =   0:  Solving these gives x =   0,   y = 175\r\n" );
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document.write( "4.  **Evaluate the objective function at each vertex:**\r\n" );
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document.write( "    *   C(175,    0)   = 8*175  + 6*0    = 1400\r\n" );
document.write( "    *   C( 87.5, 12.5) = 8*81.5 + 6*12.5 =  727\r\n" );
document.write( "    *   C(25, 75)      = 8*25   + 6*75   =  650\r\n" );
document.write( "    *   C(0, 175)      = 8*0    + 6*175  = 1050\r\n" );
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document.write( "5.  **Determine the minimum cost:** The minimum cost is the smallest value of the objective function.\r\n" );
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document.write( "The minimum cost is ₱650.\r\n" );
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document.write( "**c. Determine the best combination of ingredients:**\r\n" );
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document.write( "The best combination of ingredients is the (x, y) values that correspond to the minimum cost.\r\n" );
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document.write( "The minimum cost of ₱650 occurs when x = 25 and y = 75.    \r\n" );
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document.write( "Therefore, the best combination is **25 pounds of chemical A and 75 pounds of chemical B**.    <<<---===  ANSWER\r\n" );
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