document.write( "Question 1189985: A baseball team plays in a stadium that
\n" ); document.write( "holds 55,000 spectators. With the ticket price at $10, the
\n" ); document.write( "average attendance at recent games has been 27,000. A mar-
\n" ); document.write( "ket survey indicates that for every dollar the ticket price is
\n" ); document.write( "lowered, attendance increases by 3000.
\n" ); document.write( "(a) Find a function that models the revenue in terms of ticket
\n" ); document.write( "price. \n" ); document.write( "

Algebra.Com's Answer #821527 by ikleyn(52784) $\"\"$ $\"About$

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\n" ); document.write( "A baseball team plays in a stadium that
\n" ); document.write( "holds 55,000 spectators. With the ticket price at $10, the
\n" ); document.write( "average attendance at recent games has been 27,000. A mar-
\n" ); document.write( "ket survey indicates that for every dollar the ticket price is
\n" ); document.write( "lowered, attendance increases by 3000.
\n" ); document.write( "(a) Find a function that models the revenue in terms of ticket
\n" ); document.write( "price.
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document.write( "The problem tells us that if the ticket price is $10, then the attendance is 27000\r\n" );
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document.write( "and that ticket price change of $1 produces the attendance change of 3000 in opposite direction\r\n" );
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document.write( "    (when the price goes down, the attendance goes up, \r\n" );
document.write( "     and vice versa: when the price goes up, the attendance goes down).     (*)\r\n" );
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document.write( "In addition, it says that the attendance is a linear function of price.\r\n" );
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document.write( "It means that the slope of the plot is -3000, so we can write the attendance \r\n" );
document.write( "as a linear function of the ticket price in this form\r\n" );
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document.write( "        A(p) = 27000 - 3000*(p-10).\r\n" );
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document.write( "Indeed, this function is linear and satisfies the pointed properties (*).\r\n" );
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document.write( "Now, the revenue is the product of the ticket price by the attendance\r\n" );
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document.write( "    R(p) = p*A(p) = p*(27000 - 3000*(p-10)) = p*(27000 - 3000p + 30000) = p*(57000 - 3000p) = -3000p*2 + 57000p.\r\n" );
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document.write( "ANSWER.  Under given conditions, the revenue function is this quadratic function of the ticket price\r\n" );
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document.write( "                R(p) = -3000p*2 + 57000p.\r\n" );
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document.write( "CHECK.   At the ticket price of $10,  the revenue is  R(10) = -3000*10^2 + 57000*10 = -300000 + 570000 = 270000 dollars.\r\n" );
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document.write( "         Compare it with 27000*10 = 270000: these numbers coincide.\r\n" );
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document.write( "The given formula is valid until the attendance is not greater than the maximum capacity of the stadium of 55000 spectators.\r\n" );
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