document.write( "Question 1190154: Xin owns a business with 100 employees. He is downsizing his workforce and has determined for every two employees he terminates he can increase the pay of the remaining employees by $2000 each. He currently has 100 employees with an average pay of $50,000 each. Calculate the maximum amount his payroll could be if he begins terminating employees (i.e. find the maximum of the quadratic equation you create with the given information) \n" ); document.write( "

Algebra.Com's Answer #821721 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Xin owns a business with 100 employees. He is downsizing his workforce
\n" ); document.write( "and has determined for every two employees he terminates he can increase the pay
\n" ); document.write( "of the remaining employees by $2000 each. He currently has 100 employees
\n" ); document.write( "with an average pay of $50,000 each. Calculate the maximum amount
\n" ); document.write( "his payroll could be if he begins terminating employees
\n" ); document.write( "(i.e. find the maximum of the quadratic equation you create with the given information)
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document.write( "Starting condition is 100 employees with the average salary of $50,000.\r\n" );
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document.write( "After terminating two employees n times, the number of remaining employees is 100-2n\r\n" );
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document.write( "and the average salary of remaining employees is  (50000+2000n)  dollars.\r\n" );
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document.write( "The current payroll is then\r\n" );
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document.write( "    P(n) = (100-2n)*(50000+2000n)\r\n" );
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document.write( "and the problem wants you find the maximum of this quadratic function over n.\r\n" );
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document.write( "Since our quadratic function is the product of two linear binomials, \r\n" );
document.write( "we easily can determine the roots (the x-intercepts).  \r\n" );
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document.write( "    They are  n= 100/2 = 50           (from 100-2n = 0)  \r\n" );
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document.write( "         and  n= -50000/2000 = -25    (from 50000+2000n = 0).\r\n" );
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document.write( "The maximum of the function P(n) is located exactly at half way between 50 and -25, which is n= 12.5\r\n" );
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document.write( "But since n is an integer number, we round n to EITHER closest integer.\r\n" );
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document.write( "So, the problem has two solutions with the same maximum payroll:\r\n" );
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document.write( "    (1)  n= 12  gives  (100-2*12) = 76 remaining employees with the average salary of 50000+12*2000 = 74000 dollars\r\n" );
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document.write( "                                  with payroll 76*74000 = 5624000  dolars;\r\n" );
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document.write( "and \r\n" );
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document.write( "    (2)  n= 13  gives  (100-2*13) = 74 remaining employees with the average salary of 50000+13*2000 = 76000 dollars\r\n" );
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document.write( "                                  with payroll 74*76000 = 5624000  dolars.\r\n" );
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\n" ); document.write( "\n" ); document.write( "This problem is of the same type as those well known problems, where a seller changes the price \r
\n" ); document.write( "\n" ); document.write( "of the tickets/items in search for optimal (maximal) revenue.\r
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