document.write( "Question 118649: A theatre seats 500 people per show and is currently sold out with a ticket price of $16. A survey shows that for every $2 per ticket price increase, 25 fewer tickets will be sold.Write a function to model this situation and use this function to determine the ticket price that will result in the greatest revenue per show.
\n" ); document.write( "I tried (25x-500)(2x-16)=0
\n" ); document.write( " 50x^2-400x-1000x+8000
\n" ); document.write( " x^2-28+150
\n" ); document.write( " x=-b/a=28/2=$14
\n" ); document.write( "BUT lost several marks :( Can anyone tell me why :) \n" ); document.write( "

Algebra.Com's Answer #86785 by Edwin McCravy(20056) $\"\"$ $\"About$

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A theatre seats 500 people per show and is currently sold out with a ticket price of $16. A survey shows that for every $2 per ticket price increase, 25 fewer tickets will be sold.Write a function to model this situation and use this function to determine the ticket price that will result in the greatest revenue per show.
\n" ); document.write( "I tried (25x-500)(2x-16)=0
\n" ); document.write( "50x^2-400x-1000x+8000
\n" ); document.write( "x^2-28+150
\n" ); document.write( "x=-b/a=28/2=$14
\n" ); document.write( "BUT lost several marks :( Can anyone tell me why :)
\n" ); document.write( "

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document.write( "One of the best ways to explain a word problem is by\r\n" );
document.write( "first looking at an example using an arbitrary number,\r\n" );
document.write( "like, say, 7. For the reasoning involved in setting\r\n" );
document.write( "up an equation is the same.\r\n" );
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document.write( "Suppose there are 7 $2 increases. \r\n" );
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document.write( "Then the ticket price is figured by multiplying $2 by 7,\r\n" );
document.write( "getting $14, and adding to the original price of $16, \r\n" );
document.write( "getting $14 + 16$ or $30. \r\n" );
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document.write( "Then the number of tickets sold is gotten by multiplying\r\n" );
document.write( "7 by 25, getting 175 and subtracting from 500, getting\r\n" );
document.write( "500 - 175, or 325. \r\n" );
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document.write( "Then to get the revenue, we multiply $30 by 325 and get\r\n" );
document.write( "$9750. \r\n" );
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document.write( "So the equation for the revenue, R(7) is \r\n" );
document.write( "R(7) = $9750\r\n" );
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document.write( "---------------------- \r\n" );
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document.write( "Now let's use that same reasoning, and the same\r\n" );
document.write( "words, only this time using x instead of 7.\r\n" );
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document.write( "Suppose there are x $2 increases. \r\n" );
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document.write( "Then the ticket price is figured by \r\n" );
document.write( "multiplying $2 by x, getting $2x, and adding to the \r\n" );
document.write( "original price of $16, getting $16 + $2x or $2(8 + x). \r\n" );
document.write( "\r\n" );
document.write( "Then the number of tickets sold is gotten by multiplying\r\n" );
document.write( "x by 25, getting 25x and subtracting from 500, getting\r\n" );
document.write( "500 - 25x, or 25(20-x). \r\n" );
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document.write( "Then to get the revenue, we multiply $2(8 + x) by 25(20 - x)\r\n" );
document.write( "and get $50(8 + x)(20 - x) \r\n" );
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document.write( "So the equation for the revenue, R(x) is \r\n" );
document.write( "R(x) = $50(8 + x)(20 - x)\r\n" );
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document.write( "Now to get the maximum revenue, we multply\r\n" );
document.write( "that out\r\n" );
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document.write( "R(x) = 50(160+12x-x²)\r\n" );
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document.write( "R(x) = 8000 + 600x - 50x²\r\n" );
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document.write( "or, in descending order\r\n" );
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document.write( "R(x) = -50x² + 600x + 8000\r\n" );
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document.write( "Now we'll replace R(x) by y and\r\n" );
document.write( "get the equation\r\n" );
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document.write( "y = -50x² + 600x + 8000\r\n" );
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document.write( "The graph of that equation is this parabola:\r\n" );
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document.write( "To get the highest point on that graph, called the\r\n" );
document.write( "\"vertex\", we use the vertex formula:\r\n" );
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document.write( "For the graph of the parabola y = ax² + bx + c, \r\n" );
document.write( "the vertex is the point (,)\r\n" );
document.write( "where d = b² - 4ac, the discriminant.\r\n" );
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document.write( "For the graph of the parabola y = -50x² + 600x + 8000,\r\n" );
document.write( "we calculate d = (600)² - 4(-50)(8000) = 1960000\r\n" );
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document.write( "So the vertex is the point (,)\r\n" );
document.write( "or (,)\r\n" );
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document.write( "or (6,9800)\r\n" );
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document.write( "which means if there are 6 increases of $2, the most possible \r\n" );
document.write( "revenue of $9800 will be received.  6 increases of $2 amounts \r\n" );
document.write( "to increasing the original price of $16 per ticket by $12, \r\n" );
document.write( "making the optimum price to charge per ticket $28.\r\n" );
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document.write( "Edwin

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