document.write( "Question 1070691: A new movie theater holds 300 people. They found that when the price was $5 per ticket, the average sales would be 270 people per show. Each time they increase the price by $1, the average attendance drops by 15 people.
\n" ); document.write( "A) find an expression, using x, for the number of people that will go to each show \r
\n" ); document.write( "\n" ); document.write( "B) find the function R (x) that gives the amount of revenue the theater will make \r
\n" ); document.write( "\n" ); document.write( "C) what price should they charge in order to maximize revenue? \n" ); document.write( "

Algebra.Com's Answer #685709 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "A new movie theater holds 300 people. They found that when the price was $5 per ticket, the average sales would be 270 people
\n" ); document.write( "per show. Each time they increase the price by $1, the average attendance drops by 15 people.
\n" ); document.write( "A) find an expression, using x, for the number of people that will go to each show \r
\n" ); document.write( "\n" ); document.write( "B) find the function R (x) that gives the amount of revenue the theater will make \r
\n" ); document.write( "\n" ); document.write( "C) what price should they charge in order to maximize revenue?
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document.write( "1.  Let N(x) be this function: the number of viewers at the ticket price of x dollars.\r\n" );
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document.write( "    Then N(x) = 270 - 15(x-5).    (1)\r\n" );
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document.write( "    It is exact translation of this statement: \"Each time they increase the price by $1, the average attendance drops by 15 people.\"\r\n" );
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document.write( "    So, the question A) is answered, and the answer is: the attendance is \r\n" );
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document.write( "        N(x) = 270 - 15(x-5) = 345 - 15x.\r\n" );
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document.write( "    where x is the ticket price in dollars.\r\n" );
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document.write( "2.  The revenue R(x) is   R(x) = N(x)*x = x*(345 - 15x) = .     (2)\r\n" );
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document.write( "    See the plot below.   So, the question B) is answered, too.\r\n" );
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document.write( "3.  The maximum of the quadratic function (2) is achieved at x =  =  = 11.5.\r\n" );
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document.write( "     Thus the optimum price is 11.5 dollars per ticket.\r\n" );
document.write( "     The attendance then is N(11.5) = 270 - 15*(11.5-5) = 172.5 persons. \r\n" );
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document.write( "        Plot y = \r\n" );
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\n" ); document.write( "\n" ); document.write( "So, at given conditions, the owner of the theater can increase the ticket price to $11.50.
\n" ); document.write( "The attendance will decrease to 172 - 173 viewers, but the revenue will increase from 300*$5 = $1500 to 172*$11.50 = $1978.\r
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\n" ); document.write( "\n" ); document.write( "All questions are answered. The problem is solved.\r
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\n" ); document.write( "\n" ); document.write( "On finding the maximum/minimum of a quadratic function 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( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "For similar solved problems on maximazing revenue/profit see the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Using quadratic functions to solve problems on maximizing revenue/profit\r
\n" ); document.write( "\n" ); document.write( "in this site.\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
\n" ); document.write( "\"Finding minimum/maximum of quadratic functions\". \r
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