document.write( "Question 1058154: Sally Sethness assembles stereo equipment for resale in her shop. She offers two products, turntables and cassette players. She makes a profit of $10 on each turntable $6 on each cassette. Both must go through two steps in her shop - assembly and bench checking. A turntable take 12 hours to assemble and 4 hours to bench check. A cassette player take 4 hours assemble and 8 hours to bench check. Looking at this month's schedule, Sally sees that she has 60 assembly hours uncommitted and 40 hours of bench-checking time available. Find her best combinations of turntables and cassette player. What is the total profit on the combination you found? \n" ); document.write( "

Algebra.Com's Answer #837916 by ikleyn(52784) $\"\"$ $\"About$

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\n" ); document.write( "Sally Sethness assembles stereo equipment for resale in her shop.
\n" ); document.write( "She offers two products, turntables and cassette players.
\n" ); document.write( "She makes a profit of $10 on each turntable and $6 on each cassette.
\n" ); document.write( "Both must go through two steps in her shop—assembly and bench checking.
\n" ); document.write( "A turntable takes 12 hours to assemble and 4 hours to bench check.
\n" ); document.write( "A cassette player takes 4 hours to assemble but 8 hours to bench check.
\n" ); document.write( "Looking at this month's schedule, Sally sees that she has 60 assembly hours uncommitted
\n" ); document.write( "and 40 hours of bench-checking time available.
\n" ); document.write( "Use graphic linear programming to find her best combination of turntables and cassette players.
\n" ); document.write( "What is the total profit on the combination you found?
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document.write( "Let X be the number of turntable  and Y be the number of cassette players.\r\n" );
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document.write( "We want maximize the profit function\r\n" );
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document.write( "    P(X,Y) = 10X + 6Y\r\n" );
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document.write( "under the following restrictions\r\n" );
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document.write( "    12X + 4Y <= 60      (assembly time)\r\n" );
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document.write( "     4X + 8Y <= 40      (bench check time)\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( "    \r\n" );
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document.write( "    Plots y =   (red) and y =  (green)\r\n" );
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document.write( "It is a quadrilateral in QI with the vertices  at points P1 = (0,0), P2 = (0,5), P3 = (4,3), P4 = (5,0).\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  P(X,Y)  (profit)  at listed points\r\n" );
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document.write( "    P1:  P(0,0) = 10*0 + 6*0 =    0,\r\n" );
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document.write( "    P2:  P(0,4) = 10*0 + 6*5 =   30,\r\n" );
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document.write( "    P3:  P(4,3) = 10*4 + 6*3 =   58,\r\n" );
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document.write( "    P4:  P(6,0) = 10*5 + 6*0 =   50.\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  P3:  (X,Y) = (4,3),  which means  4 turntables and 3 cassette players, providing maximum PROFIT of 58 dollars.\r\n" );
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\n" ); document.write( "\n" ); document.write( "If you want to see many other similar and different 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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