document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1136921>Question 1136921</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A baseball team plays in a stadium that holds 68000 spectators. With the ticket price at $11 the average attendance has been 28000. When the price dropped to $9, the average attendance rose to 34000. Assume that attendance is linearly related to ticket price.\r<BR>\n" );
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document.write( "What ticket price would maximize revenue? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #754776 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=754776\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> when the price is 11, the attendance is 28000.<BR>\n" );
document.write( "when the price is 9, the attendance is 34000.\r<BR>\n" );
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document.write( "when the price drops by 2 dollars, the attendance goes up by 6000.\r<BR>\n" );
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document.write( "since this is linear, the slope intercept form of the equation would be y = mx + b.\r<BR>\n" );
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document.write( "m is the slope and b is the y-intercept.\r<BR>\n" );
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document.write( "the slope is (y2 - y1) / (x2 - x1).\r<BR>\n" );
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document.write( "(x1,y1) = (11,28000)<BR>\n" );
document.write( "(x2,y2) = (9,34000)\r<BR>\n" );
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document.write( "the slope is (y2 - y1) / (x2 - x1) = (34000 - 28000) / (9 - 11) = 6000 / -2 = -3000.\r<BR>\n" );
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document.write( "y = mx + b becomes y = -3000 * x + b\r<BR>\n" );
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document.write( "solve for b by replacing x and y with one of the points.\r<BR>\n" );
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document.write( "i chose (9,34000).\r<BR>\n" );
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document.write( "y = -3000 * x + b becomes 34000 = -3000 * 9 + b\r<BR>\n" );
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document.write( "solve for b to get b = 34000 + 27000 = 61000\r<BR>\n" );
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document.write( "the slope intercept form of the equation for the price is y = -3000 * x + 61000.\r<BR>\n" );
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document.write( "this equation can be graphed and is shown below.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/031901.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the revenue equation is y = (11 - x) * (28000 + 3000 * x)\r<BR>\n" );
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document.write( "that equation can be graphed and is shown below.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/031902.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the revenue equation is a quadratic equation.\r<BR>\n" );
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document.write( "the standard form of that equation is y = -3000 * x^2 + 5000 * x + 308000.\r<BR>\n" );
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document.write( "in standard form, you get a = -3000, b = 5000, c = 308000.\r<BR>\n" );
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document.write( "the maximum revenue occurs when x = -b / 2a.\r<BR>\n" );
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document.write( "that becomes -5000 / -6000 = 5 / 6 = .8333333333.....\r<BR>\n" );
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document.write( "when x = .833333....., price = 11 - .8333333 = 10.666666666 and quantity becomes 28000 + 3000 * .8333333 = 30500.\r<BR>\n" );
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document.write( "since revenue = price * quantity, revenue = 10.166666666... * 30500 = 310083.33333333....\r<BR>\n" );
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document.write( "the maximum revenue can be seen on the following graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/031903.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "you have two basic equations associated with this problem.\r<BR>\n" );
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document.write( "the first equation is y = -3000 * x + 61000.\r<BR>\n" );
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document.write( "in this equation, x is the price and y is the quantity of tickets sold.\r<BR>\n" );
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document.write( "when the price is 0, you can sell 61000 tickets.\r<BR>\n" );
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document.write( "notice that the capacity is 68000, so you won't fill up the stadium even if you give the tickets away, based on this equation.\r<BR>\n" );
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document.write( "when the price is 20 and 1/3 per ticket. you will sell none based on this equation.\r<BR>\n" );
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document.write( "when the price is 11, you sell 28000.\r<BR>\n" );
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document.write( "when the price is 9, you sell 34000.\r<BR>\n" );
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document.write( "the second equation is y = (11 - x) * (28000 + 3000 * x)\r<BR>\n" );
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document.write( "in this equation, y is the revenue and x is the change in the price and 3000 * x is the change in the quantity for each change in the price.\r<BR>\n" );
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document.write( "when x = 0, the price is 11 and the quantity is 28000 and the revenue is 11 * 28000 = 308000.\r<BR>\n" );
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document.write( "when x = 2, the price is 9 and the quantity is 34000 and the revenue is 9 * 34000 = 306000.\r<BR>\n" );
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document.write( "when x is 11, the price is 0 and the quantity is 61000 and the revenue is 0 * 61000 = 0.\r<BR>\n" );
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document.write( "when x is -9.333333333, the price is 20.33333333 and the quantity is 0 and the revenue = 0.\r<BR>\n" );
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document.write( "the same revenue graph also shows when the maximum revenue occurs.\r<BR>\n" );
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document.write( "that occurs when x = .8333333333.\r<BR>\n" );
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document.write( "the price is 11 - .8333333333 = 10.1666666666... and the quantity is 28000 + .8333333333 * 3000 = 30500 and the revenue is 10.166666..... * 30500 = 310083.33333.....\r<BR>\n" );
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document.write( "the maximum revenue is shown on the last graph.\r<BR>\n" );
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document.write( "it's the same as the second graph except only the maximum revenue is shown for clarity.\r<BR>\n" );
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document.write( "note that the equation shown on the revenue graph is y = (11 - x) * (28000 + 3000 * x).\r<BR>\n" );
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document.write( "this is the same equation as y = -3000 * x^2 + 5000 * x +308000.\r<BR>\n" );
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document.write( "those two equations are equivalent to each other, just shown in different forms.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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