document.write( "Question 1177940: A manufacturer of downhill and cross-country skis reports that manufacturing time is 3 hours and 7 hours, respectively, per ski and that finishing time is 2 hours for each downhill and 2 hours for each cross-crountry ski. There are only 35 hours per week available for the manufacturing process and 18 hours for the finishing process. The average profit is $62 for downhill ski and $63 for cross-country ski. The manufacturer wants to know how many of each type of ski should be made to maximize the weekly profit. \r
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Algebra.Com's Answer #807118 by ikleyn(52786)\"\" \"About 
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\n" ); document.write( "A manufacturer of downhill and cross-country skis reports that manufacturing time is 3 hours and 7 hours, respectively, per ski
\n" ); document.write( "and that finishing time is 2 hours for each downhill and 2 hours for each cross-crountry ski.
\n" ); document.write( "There are only 35 hours per week available for the manufacturing process and 18 hours for the finishing process.
\n" ); document.write( "The average profit is $62 for downhill ski and $63 for cross-country ski.
\n" ); document.write( "The manufacturer wants to know how many of each type of ski should be made to maximize the weekly profit.
\n" ); document.write( "What are the corner points of the feasible region?
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document.write( "Let X be the number of downhill skis, and\r\n" );
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document.write( "let Y be the number of cross-country skis.\r\n" );
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document.write( "Then the manufacturing time to produce X downhill skis and Y cross-country skis is  3X + 7Y  hours.\r\n" );
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document.write( "and  the finishing     time to produce X downhill skis and Y cross-country skis is  2X + 2Y  hours.\r\n" );
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document.write( "Therefore, the two major constraints are\r\n" );
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document.write( "    3X + 7Y <= 35 hours    (manufacturing time)     (1)\r\n" );
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document.write( "    2X + 2Y <= 18 hours    (finishing     time)     (2)\r\n" );
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document.write( "Add to it the standard non-negativity constraints\r\n" );
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document.write( "    X >= 0,  Y >= 0.                                (3)\r\n" );
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document.write( "Now we have everything to construct the feasibility region.\r\n" );
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document.write( "For it, you draw the lines  3X + 7Y = 35  and  2X + 2Y = 18  from the constraints.\r\n" );
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document.write( "They are shown in the figure below.\r\n" );
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document.write( "    \r\n" );
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document.write( "                Plot y =  \"%2835-3x%29%2F7\" (red)  and  y = \"%2818-2x%29%2F2\" (green)\r\n" );
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document.write( "Inequalities (3) define the first quadrant QI.\r\n" );
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document.write( "The solution set to the given inequalities (the feasibility domain) are the points of the coordinate plane, \r\n" );
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document.write( "    that are in QI and belong the quadrilateral below (or on) the red line and below (or on) the green line.\r\n" );
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document.write( "Again: the feasibility domain is the quadrilateral in QI restricted by the red and by the green lines, \r\n" );
document.write( "adjacent to coordinate axes, including its sides and vertices.\r\n" );
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document.write( "The corner points are\r\n" );
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document.write( "    1)  P1 = (0,0)   the origin of the coordinate plane\r\n" );
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document.write( "    2)  P2 = (0,5)   Y-intercept of the red line       \r\n" );
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document.write( "    3)  P3 = (7,2)   intersection of the red line and the green line \r\n" );
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document.write( "    4)  P4 = (9,0)   X-intercept of the green line  \r\n" );
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document.write( "At this point, I completed my explanations regarding the corner points.\r\n" );
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document.write( "Next, to solve this minimax problem,  compare the values of the objective function z = 62X + 63Y at the corner points.\r\n" );
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document.write( "    P1:  z = 62*0 + 63*0 = 0\r\n" );
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document.write( "    P2:  z = 62*0 + 63*5 = 315\r\n" );
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document.write( "    P3:  z = 62*7 + 63*2 = 560\r\n" );
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document.write( "    P4:  z = 62*9 + 63*0 = 558\r\n" );
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document.write( "You see that the profit is maximum at point P3.\r\n" );
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document.write( "It gives the solution to the problem : X= 7 downhill skis,  Y= 2 cross-country skis with the maximum profit of 560 dollars.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "To see many other similar solved problems, look into the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Solving minimax problems by the Linear Programming method \r
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