document.write( "Question 1119561: A graphic designer can design a magazine cover or a logo. Her company makes a profit of $800 for each magazine cover and $500 for each logo. She estimates that it takes her 4 hours of brainstorming for a magazine cover and 2 hours of brainstorming for a logo. She'd like to keep the total brainstorming time under 24 hours a week. Further, she estimates that it takes her 2 hours to lay out a magazine cover and 0.5 hours to sketch up a logo, and she must fit this into 10 hours a week. Her boss requires her to design no more than 4 logos for each magazine cover she designs.\r
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Algebra.Com's Answer #735128 by ikleyn(52790) $\"\"$ $\"About$

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document.write( "Let X = the number of the magazine covers,\r\n" );
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document.write( "    Y = the number of logos. \r\n" );
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document.write( "Then the profit function is\r\n" );
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document.write( "P(X,Y) = 800*X + 500*Y      (1)    dollars.\r\n" );
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document.write( "The restrictions are \r\n" );
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document.write( "4*X + 2*Y <= 24       (2)    (hours per week)\r\n" );
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document.write( "2*X + 0.5*Y <= 10     (3)    (hours per week)\r\n" );
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document.write( "Y <= 4X               (4)    (\"no more than 4 logos for each magazine cover\")\r\n" );
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document.write( "Other restrictions are non-negativity\r\n" );
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document.write( "X >= 0;  Y >= 0.\r\n" );
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document.write( "The feasible domain is shown in the plot below.\r\n" );
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document.write( "    \r\n" );
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document.write( "    Plot  4*X + 2*Y = 24 (red);  2*X + 0.5*Y = 10  (green);  and  Y = 4X (blue)\r\n" );
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document.write( "It is the quadrilateral in QI, adjacent to x-axis and bounded by the blue, red and green lines.\r\n" );
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document.write( "It has vertices \r\n" );
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document.write( "    P1 = (2,8)       \r\n" );
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document.write( "    P2 = (4,4)    and\r\n" );
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document.write( "    P3 = (5,0)       (the green line x-intercept).\r\n" );
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document.write( "According to the Linear Programming method, we should calculate and compare the values of the profit function at these three points\r\n" );
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document.write( "    at P1:  P(2,8) = 800*2 + 500*8 = 5560 dollars;\r\n" );
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document.write( "    at P2:  P(4,4) = 800*4 + 500*4 = 5200 dollars.\r\n" );
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document.write( "    at P3:  P(5,0) = 800*5 + 500*0 = 4000 dollars.\r\n" );
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document.write( "The maximum value of the profit function is at P1.\r\n" );
document.write( "It is the optimal solution.\r\n" );
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document.write( "Answer.  Optimal solution to the problem is 4 magazine covers and 8 logos.\r\n" );
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document.write( "         It satisfies the restrictions and gives maximal profit of 5560 dollars.\r\n" );
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\n" ); document.write( "\n" ); document.write( "To see other similar problems solved by the Linear Programming method, look into the lesson\r
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