document.write( "Question 1085911: A farmer has 120 acres of land on which she plans to grow wheat and corn. Each acre of wheat requires 5 hours of labor and $20 of capital, and each acre of corn requires 15 hours of labor and $30 of capital. The farmer has at most 1500 hours of labor and $6000 of capital available. If the profit from an acre of wheat is $100 and from an acre of corn is $120, how many acres of each crop should she plant to maximize her profit? \n" ); document.write( "

Algebra.Com's Answer #700065 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "A farmer has 120 acres of land on which she plans to grow wheat and corn.
\n" ); document.write( "Each acre of wheat requires 5 hours of labor and $20 of capital,
\n" ); document.write( "and each acre of corn requires 15 hours of labor and $30 of capital.
\n" ); document.write( "The farmer has at most 1500 hours of labor and $6000 of capital available. \r
\n" ); document.write( "\n" ); document.write( "If the profit from an acre of wheat is $100 and from an acre of corn is $120,
\n" ); document.write( "how many acres of each crop should she plant to maximize her profit?
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document.write( "Let X be the area for wheat (in acres),  and\r\n" );
document.write( "let Y be the area for corn.\r\n" );
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document.write( "The amount of labor is 5X + 15Y     hours.\r\n" );
document.write( "The amount of capital is 20X + 30Y  dollars.\r\n" );
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document.write( "The constraints are \r\n" );
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document.write( " 5x + 15Y <= 1500     (1)    (hours)\r\n" );
document.write( "20X + 30Y <= 6000     (2)    (dollars)\r\n" );
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document.write( "X >= 0,               (3)\r\n" );
document.write( "Y >= 0.               (4)\r\n" );
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document.write( "The profit function is P(X,Y) = 100X + 120Y.   (5)\r\n" );
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document.write( "So, the problem is to maximize (5) under restrictions (1) - (4). \r\n" );
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document.write( "The feasibility area (1) - (4) is shown in the figure below.\r\n" );
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document.write( "Plots y =  (red)  and y =  (green)\r\n" );
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document.write( "The feasibility area is the intersection of two right-angled triangles: one with red hypotenuse (restrictions (1),(3),(4))\r\n" );
document.write( "and the other with green hypotenuse (restrictions (2),(3),(4)).\r\n" );
document.write( "The intersection is the smaller right-angled triangle,\r\n" );
document.write( "and to get the solution of the minimax problems, you should compare the profit function at two points: (X,Y) = (0,100) and (X,Y) = (300,0).\r\n" );
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document.write( "P(0,100) = 100*0 + 120*100 = 12000 dollars,\r\n" );
document.write( "P(300,0) = 100*300 + 120*0 = 30000 dollars.\r\n" );
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document.write( "The larger value is at the point (X,Y) = (300,0).\r\n" );
document.write( "So the solution is (X,Y) = (300,0), i.e. 300 acres for wheat and 0 acres for corn.\r\n" );
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document.write( "The profit value expected/predicted is $30000.\r\n" );
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\n" ); document.write( "\n" ); document.write( "To see more minimax problems/samples solved, look into the lesson\r
\n" ); document.write( "\n" ); document.write( " - Solving minimax problems by the Linear Programming method \r
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