document.write( "Question 1146167: When a school play charges 2$ for admission, an average of 100 people attend. For each 10 cents increase in admission price, the average number decreases by 1. What charge would make the most money? \n" ); document.write( "

Algebra.Com's Answer #768015 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "The meaning of this problem is that when the admission price varies as p = 2+0.1x dollars, \r\n" );
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document.write( "the number of people who attend varies as n(x) = 100 - x.\r\n" );
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document.write( "The revenue function then is\r\n" );
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document.write( "    R(x) = n*p = (100-x)*(2+0.1x)      (1)\r\n" );
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document.write( "It is a quadratic function with negative leading coefficient.\r\n" );
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document.write( "They want you find an admission price which provides the maximum value to the revenue.\r\n" );
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document.write( "For it, you need to find the value of \"x\" which provides the maximum to the quadratic function (1), and then to calculate the price 2 + 0.1x.\r\n" );
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document.write( "This value of x is exactly half way between the roots of linear binomials 100-x and 2+0.1x.\r\n" );
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document.write( "The roots are  x= 100  and  x= -20;  hence, the value of x which provides the maximum value is   =  = 40.\r\n" );
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document.write( "Then the optimum price is  2 + 40*0.1 = 2 + 4 = 6 dollars for admission.\r\n" );
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document.write( "The attendance then is  p(40) = 100 - 40 = 60  and the revenue is  60*6 = 360 dollars.\r\n" );
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document.write( "Compare it with the revenue of 2*100 = 200 dollars before the optimization.\r\n" );
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document.write( "The plot of the revenue as the function (1) is below\r\n" );
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document.write( "    \r\n" );
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document.write( "    Plot y = (100-x)*(2+0.1x)\r\n" );
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\n" ); document.write( "\n" ); document.write( "On finding the maximum/minimum of quadratic functions and relevant solved problems, see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the vertex of a parabola\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the vertex of a parabola\r
\n" ); document.write( "\n" ); document.write( "    - A rectangle with a given perimeter which has the maximal area is a square\r
\n" ); document.write( "\n" ); document.write( "    - A farmer planning to fence a rectangular garden to enclose the maximal area\r
\n" ); document.write( "\n" ); document.write( "    - A farmer planning to fence a rectangular area along the river to enclose the maximal area\r
\n" ); document.write( "\n" ); document.write( "    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area\r
\n" ); document.write( "\n" ); document.write( "    - Using quadratic functions to solve problems on maximizing revenue/profit (*)\r
\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of lessons on finding the maximum/minimum of a quadratic function\r
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\n" ); document.write( "\n" ); document.write( "and especially the lesson marked (*) in the list.\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( "Save the link to this online textbook together with its description\r
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\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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