document.write( "Question 1118425: A manufacturer produces two models of mountain bikes. Model A requires 5 hours of assembly time and 2 hours of painting time, and Model B requires 4 hours of assembly time and 3 hours of painting time. The maximum total weekly hours available in the assembly department and the painting department are 200 hours and 108 hours, respectively. The profits per unit are $25 for Model A and $15 for Model B. How many of each type should be produced to maximize profit?\r
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Algebra.Com's Answer #733798 by ikleyn(52794)\"\" \"About 
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\n" ); document.write( "A manufacturer produces two models of mountain bikes.
\n" ); document.write( "Model A requires 5 hours of assembly time and 2 hours of painting time,
\n" ); document.write( "and Model B requires 4 hours of assembly time and 3 hours of painting time.
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\n" ); document.write( "200 hours and 108 hours, respectively.
\n" ); document.write( "The profits per unit are $25 for Model A and $15 for Model B. How many of each type should be produced to maximize profit?
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document.write( "Let X be the number of bikes Model A;\r\n" );
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document.write( "    Y be the number of bikes Model B.\r\n" );
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document.write( "Then the constraints are \r\n" );
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document.write( "5X + 4Y <= 200,         (1)     (Assembly time constraint of 200 hours)\r\n" );
document.write( "2X + 3Y <= 108.         (2)     (Painting time constraint of 108 hours)\r\n" );
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document.write( "X >= 0, Y >= 0.         (3)     (Non-negativity constraint)\r\n" );
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document.write( "The profit function is\r\n" );
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document.write( "P(X,Y) = 25X + 15Y.     (4)\r\n" );
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document.write( "The feasibility domain is shown in the Figure below.\r\n" );
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document.write( "It is the quadrilateral in the Quadrant I below two straight lines that are constraints.\r\n" );
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document.write( "Plots  5x + 4y = 200 (red line)  and  2x + 3y = 108 (green line).\r\n" );
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document.write( "The vertices of  the quadrilateral are  \r\n" );
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document.write( "    P1 = (0,36)    (y-intersept of the green line)\r\n" );
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document.write( "    P2 = (40,0)    (x-intercept of the red line)\r\n" );
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document.write( "    P3 = (24,20)   (the intersection point of the red and green lines)   <<<---=== You may find it by solving the system of equations\r\n" );
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document.write( "    P4 = (0,0)    (the origin of the coordinate system)\r\n" );
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document.write( "According to the Linear Programming method, you need to find the values of the profit function in the vertices of the feasibility domain,\r\n" );
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document.write( "and then select the vertex, where the value of the profit function is maximal.\r\n" );
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document.write( "These calculations re shown below:\r\n" );
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document.write( "    at P1:   P(X,Y) = P(0,36)  = 0*25  + 36*15 =  540;\r\n" );
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document.write( "    at P2:   P(X,Y) = P(40,0)  = 40*25 + 0*15  = 1000;\r\n" );
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document.write( "    at P3:   P(X,Y) = P(24,20) = 24*25 + 20*15 = 900;\r\n" );
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document.write( "    at P4:   P(X,Y) = P(0,0)   = 0*25  + 0*15  =   0.\r\n" );
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document.write( "The maximum value is 1000 at P2;  so,  X = 40, Y = 0 is the solution to the given problem.\r\n" );
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document.write( "Answer.  40 bikes of the model A  and  0 (zero, ZERO) bikes of the model B  satisfy the constraints \r\n" );
document.write( "and provide the maximum profit of 1000 dollars.\r\n" );
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