document.write( "Question 1183433: A manufacturer has found that if his product is priced at N90/unit then his weekly demand is 50 units, but the demand rises to 60 units per week if the selling price is N70/unit. His weekly fixed cost is N3, 000 and variable cost N20/unit.
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Algebra.Com's Answer #813773 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "A manufacturer has found that if his product is priced at N90/unit then his weekly demand is 50 units,
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document.write( "Let \"n\" be the number of units produced and sold.\r\n" );
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document.write( "Then the total cost function is  C(n) = 3000 + 20n.\r\n" );
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document.write( "Now we want to determine the revenue function, and for it, we need to know the unit selling price as the function of production.\r\n" );
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document.write( "The problem tells us that the unit selling price falls N20 from N90 to N70  when production rises  10 units from 50 to 60.\r\n" );
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document.write( "We assume (as always in such problems), that the unit selling price function  p(n) is linear, so it is  p(n) = 90 - 2(n-50).\r\n" );
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document.write( "Therefore, the revenue is\r\n" );
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document.write( "    R(n) = n*p(n) = n*(90-2(n-50)) = 90n -2n^2 + 100n = -2n^ + 190n.\r\n" );
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document.write( "Now the profit is\r\n" );
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document.write( "    P(n) = R(n) - C(n) = -2n^2 + 190n - 3000 - 20n = -2n^2 + 170n - 3000.    (1)\r\n" );
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document.write( "The problem asks about the maximum of the profit, which is the vertex of the quadratic function (1).\r\n" );
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document.write( "The maximum happens at  n = \"  \", which is   =  = 42.5.\r\n" );
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document.write( "The number of units must be integer, so the solution are two integer numbers closest to 42.5, i.e. 42 and 43.\r\n" );
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document.write( "To find the maximum profit, substitute this found value n= 42 into the profit function (1)\r\n" );
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document.write( "    P(42) = -2*42^2 + 170*42 - 3000 = 612.\r\n" );
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document.write( "ANSWER.  The optimal production level is of 42 or 43 units generating the maximum profit of N612.\r\n" );
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document.write( "        The plot to provide visual check.\r\n" );
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document.write( "     Plot R(n) = -2n^2 + 170n - 3000 (red) and the plot y = 612 (green)\r\n" );
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