document.write( "Question 1122966: A real estate company owns 228 efficiency apartments, which are fully occupied when the rent is $990 per month. The company estimates that for each $25 increase in rent, 5 apartments will become unoccupied. What rent should be charged so that the company will receive the maximum monthly income? \n" ); document.write( "

Algebra.Com's Answer #739151 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "The condition says \"for each $25 increase in rent, 5 apartments will become unoccupied\".\r\n" );
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document.write( "It is the same as (is equivalent) to say that for each $5 increase in rent, 1 apartment will become unoccupied.\r\n" );
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document.write( "So inter-relation between the number of occupied apartments \"n\" and the rent cost \"p\" (payment, in dollars) is THIS :\r\n" );
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document.write( "    if the price is  p(m) = 990 + 5m,  then the number of occupied apartments is n = 228 - m,  where \"m\" is an integer number (a parameter).\r\n" );
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document.write( "It means that the revenue is\r\n" );
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document.write( "    R(m) = p*n = p(m)*n(m) = (990 + 5m)*(228-m).\r\n" );
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document.write( "It is a quadratic function of m.  It is more convenient to present it as the quadratic function of the real argument x\r\n" );
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document.write( "    R(x) = {990 + 5x)*(228-x).\r\n" );
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document.write( "This quadratic function, obviously, is open downward (has negative coefficient at x^2); so, it has maximum, and our goal is to find this maximum.\r\n" );
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document.write( "Note that the roots of this quadratic polynomial are easy to find by equating each factor to zero.\r\n" );
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document.write( "It gives the zeroes  x =  = -198  and  x = 228.\r\n" );
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document.write( "The maximum of the quadratic function is achieved exactly mid-way between the zeroes - so the maximum is at  x=  = 15.\r\n" );
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document.write( "It means that the optimal price is  p(m) = p(15) = 990 + 5m = 990 + 5*15 = 1065 dollars,\r\n" );
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document.write( "which provides the number of occupied apartments  n(m) = n(15) = 228 - m = 228 - 15 = 213.\r\n" );
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document.write( "Then the revenue is  213*1065 =  226845 dollars.\r\n" );
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document.write( "Compare it with the revenue at full occupancy:  228*990 = 225720 dollars.\r\n" );
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\n" ); document.write( "\n" ); document.write( "There is a special lesson in this site\r
\n" ); document.write( "\n" ); document.write( " - Using quadratic functions to solve problems on maximizing revenue/profit\r
\n" ); document.write( "\n" ); document.write( "where you can find the solutions to many other similar problems.\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( " - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this textbook under the topic \"Finding minimum/maximum of quadratic functions\". \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
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