document.write( "Question 1066773: Carlo and Anita make mailboxes and toys in their craft shop near Lincoln. Each mailbox requires 3 hours of work from Carlo and 4 hours from Anita. Each toy requires 2 hours of work from Carlo and 4 hours from Anita. Carlo cannot work more than 18 hours per week and Anita cannot work more than 32 hours per week. If each mailbox sells for $11 and each toy sells for $12, then how many of each should they make to maximize their revenue? What is their maximum revenue? \n" ); document.write( "

Algebra.Com's Answer #681998 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Carlo and Anita make mailboxes and toys in their craft shop near Lincoln.
\n" ); document.write( "Each mailbox requires 3 hours of work from Carlo and 4 hours from Anita.
\n" ); document.write( "Each toy requires 2 hours of work from Carlo and 4 hours from Anita.
\n" ); document.write( "Carlo cannot work more than 18 hours per week and
\n" ); document.write( "Anita cannot work more than 32 hours per week.
\n" ); document.write( "If each mailbox sells for $11 and each toy sells for $12,
\n" ); document.write( "then how many of each should they make to maximize their revenue? What is their maximum revenue?
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document.write( "The question is: how many mailboxes (X) and how many toys (Y) should be produced to maximize the revenue $11*X + $12*Y\r\n" );
document.write( "under these restrictions:\r\n" );
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document.write( "3X + 2Y <= 18     (1)     (Carlo restricted by 18 hours per week) and\r\n" );
document.write( "4X + 4Y <= 32     (2)     (Anita restricted by 32 hours per week).\r\n" );
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document.write( "In other words, you must maximize the objective function (revenue) \r\n" );
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document.write( "R(X,Y) = 11X + 12Y\r\n" );
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document.write( "over the domain on the plot below, which is  a quadrilateral in the first quadrant (X >= 0,  Y >= 0) restricted \r\n" );
document.write( "by the red and the green lines.\r\n" );
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document.write( "Plots y =   (red) and y =  (green)\r\n" );
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document.write( "The method of linear programming says:\r\n" );
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document.write( "    1) Take the vertices of this quadrilateral\r\n" );
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document.write( "        (x1,Y1) = (0,8)   (green line Y-intercept)\r\n" );
document.write( "        (x2,Y2) = (6,0)   (red line X-intercept)\r\n" );
document.write( "        (x3,Y3) = (2,6)   (intersection point of the straight lines Y =  and Y =  )\r\n" );
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document.write( "    2) Calculate the objective function at these points\r\n" );
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document.write( "        R(X1,Y1) = 11*0 + 12*8 = 96;\r\n" );
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document.write( "        R(X2,Y2) = 11*6 + 12*0 = 66;\r\n" );
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document.write( "        R(X3,Y3) = 11*2 + 12*6 = 94.\r\n" );
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document.write( "    3) Then select one of these point where the objective function is maximal - In our case this point is (X1,Y1) = (0,8)\r\n" );
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document.write( "    4) This point gives your optimal solution X = 0 mailboxes and Y = 8 toys.\r\n" );
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document.write( "If they follow this optimal solution, their weekly revenue will be MAXIMAL, $96.\r\n" );
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