document.write( "Question 1185076: The Halloween Express is making pre-packaged bags of candy for customers to give out to trick-or-treaters. The store has 20 tootsie rolls and 18 suckers. Package A requires 1 tootsie roll and 3 suckers. Package B requires 5 tootsie rolls and 1 sucker. If package A sells for $2 and package B sells for $3, how many of each package should they make to maximize profit? \n" ); document.write( "

Algebra.Com's Answer #815816 by ikleyn(52787) $\"\"$ $\"About$

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\n" ); document.write( "The Halloween Express is making pre-packaged bags of candy for customers to give out to trick-or-treaters.
\n" ); document.write( "The store has 20 tootsie rolls and 18 suckers.
\n" ); document.write( "Package A requires 1 tootsie roll and 3 suckers.
\n" ); document.write( "Package B requires 5 tootsie rolls and 1 sucker.
\n" ); document.write( "If package A sells for $2 and package B sells for $3, how many of each package should they make to maximize $\"cross%28profit%29\"$ revenue ?
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\n" ); document.write( "\n" ); document.write( "            It is quite obvious, that at the given circumstances, the problem's question
\n" ); document.write( "            should be about the REVENUE and not about the profit.\r
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\n" ); document.write( "\n" ); document.write( "            With this editing, see my solution below.\r
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document.write( "Let X be the number of packages A  and Y be the number of packages B.\r\n" );
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document.write( "Then we want maximize the revenue function\r\n" );
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document.write( "    R(X,Y) = 2X + 3Y\r\n" );
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document.write( "under the following restrictions\r\n" );
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document.write( "     X + 5Y <= 20      (toorsie rolls)\r\n" );
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document.write( "    3X +  Y <= 18      (suckers)\r\n" );
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document.write( "     X >= 0,  Y >= 0\r\n" );
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document.write( "The plot of the feasibility domain is shown in the Figure below.\r\n" );
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document.write( "    Plots y =   (red) and y = 18-3x (green)\r\n" );
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document.write( "It is a quadrilateral in QI with the vertices  (X,Y) = (0,0), (0,4), (5,3), (6,0).\r\n" );
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document.write( "We apply the standard Linear Programming method in its geometric interpretation.\r\n" );
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document.write( "The solution is one of these 4 points, where the objective function (profit) has a maximum.\r\n" );
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document.write( "You calculate the values of the objective function  R(X,Y)  (revenue)  at listed points\r\n" );
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document.write( "    P(0,0) = 2*0 + 3*0 =    0\r\n" );
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document.write( "    P(0,4) = 2*0 + 3*4 =   12\r\n" );
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document.write( "    P(5,3) = 2*5 + 3*3 =   19\r\n" );
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document.write( "    P(6,0) = 2*6 + 3*0 =   12.\r\n" );
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document.write( "Comparing these values, you find the optimal point.\r\n" );
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document.write( "It is  (X,Y) = (5,3),  which means  5 boxes A and 3 boxes B, providing maximum REVENUE of 19 dollars.\r\n" );
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\n" ); document.write( "\n" ); document.write( "In this site, there is a lesson\r
\n" ); document.write( "\n" ); document.write( " - Solving minimax problems by the Linear Programming method \r
\n" ); document.write( "\n" ); document.write( "which explains, for beginners, metodology of solving such problems in more details.\r
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