document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1002286>Question 1002286</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A city's transit authority serves 192,000 commuters daily when fair is $1.70. Market research has determined that every penny decrease in the fare will result in 1200 new riders. What fare will maximize revenue?\r<BR>\n" );
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document.write( "We are working on Quadratic Functions, and word problems are especially difficult for me to figure out. This is what I have and the answer is incorrect:\r<BR>\n" );
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document.write( "Revenue = Price(number sold)<BR>\n" );
document.write( "  Revenue = y\r<BR>\n" );
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document.write( "y=P(1.70-P)\r<BR>\n" );
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document.write( "y= 1.70p-p^2\r<BR>\n" );
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document.write( "using the formula for a Vertex of a Parabola: (-b/2a, c-b^2/4a)\r<BR>\n" );
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document.write( "a= -1    b= 1.70   c= 0\r<BR>\n" );
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document.write( "using the \"x\" part of that formula:\r<BR>\n" );
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document.write( "x= -1.70/2(-1)\r<BR>\n" );
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document.write( "I got the answer of $0.85  (WHICH IS INCORRECT)\r<BR>\n" );
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document.write( "Thank you<BR>\n" );
document.write( "Angy </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #619235 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=619235\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> this one is particularly difficult to set up so don't feel too bad.\r<BR>\n" );
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document.write( "i haven't done it in a while and i'm struggling to set it up properly.\r<BR>\n" );
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document.write( "start with r = n * p\r<BR>\n" );
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document.write( "r = revenue<BR>\n" );
document.write( "n = number of tickets sold<BR>\n" );
document.write( "p = price per ticket.\r<BR>\n" );
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document.write( "you know that, when p = 1.71, n = 192000\r<BR>\n" );
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document.write( "you get r = 192000 * 1.71 = 328320\r<BR>\n" );
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document.write( "you also know that, when you drop the price by .01, n goes up 1200.\r<BR>\n" );
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document.write( "let 192000 + 1200 * x = the increase in n for every drop in price of .01\r<BR>\n" );
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document.write( "let 1.71 - .01 * x  = the drop in price of .01.\r<BR>\n" );
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document.write( "every time x increases by 1, the price will drop by .01<BR>\n" );
document.write( "every time x increases by 1, the number of tickets sold will go up 1200.\r<BR>\n" );
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document.write( "your formula is:\r<BR>\n" );
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document.write( "y = (192000 + 1200 * x) * (1.71 - .01 * x)\r<BR>\n" );
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document.write( "y is the revenue<BR>\n" );
document.write( "(192000 + 1200 * x) is the number of tickets sold.<BR>\n" );
document.write( "(1.71 - .01 * x) is the price of each ticket.\r<BR>\n" );
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document.write( "if x is 0, then you get:\r<BR>\n" );
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document.write( "y = (192000 + 1200 * x) * (1.71 - .01 * x) becomes:<BR>\n" );
document.write( "y = (192000 + 1200 * 0) * (1.71 - .01 * 0) which becomes:<BR>\n" );
document.write( "y = 192000 * 1.71 = 328320\r<BR>\n" );
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document.write( "any time x increases by .01, the number of tickets sold increases by 1200.<BR>\n" );
document.write( "this is because the price has decreased by .01.\r<BR>\n" );
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document.write( "any time x decreases by .01, the number of tickets sold decreases by 1200.<BR>\n" );
document.write( "this is because the price has increased by .01.\r<BR>\n" );
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document.write( "your maximum point will be when x = 5.5\r<BR>\n" );
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document.write( "you will get:\r<BR>\n" );
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document.write( "y = (192000 + 1200 * 5.5) (1.71 - .01 * 5.5) which results in:\r<BR>\n" );
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document.write( "y = 328683.\r<BR>\n" );
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document.write( "that's your maximum revenue.\r<BR>\n" );
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document.write( "simplify the equation by multiplying the factors to get:\r<BR>\n" );
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document.write( "y = (192000 + 1200 * x) * (1.71 - .01 * x) becomes:\r<BR>\n" );
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document.write( "y = (192000 * 1.71) - (192000 * .01 * x) + (1200 * x * 1.71) - (1200 * x * .01 * x)\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "y = 328320 - (1920 * x) + 2052 * x) - (12 * x^2)\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "y = 328320 + 132 * x - 12 * x^2\r<BR>\n" );
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document.write( "re-order terms in descending order of degree to get:\r<BR>\n" );
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document.write( "y = -12x^2 + 132x + 328320\r<BR>\n" );
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document.write( "now that your equation is in standard quadratic form, you can solve for maximum point as follows:\r<BR>\n" );
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document.write( "a = -12<BR>\n" );
document.write( "b = 132<BR>\n" );
document.write( "c = 328320\r<BR>\n" );
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document.write( "x = -b/2a becomes x = -132 / -24 which becomes x = 5.5\r<BR>\n" );
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document.write( "when x = 5.5, the equation of y = -12x^2 + 132x + 328320 becomes:\r<BR>\n" );
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document.write( "y = -12(5.5)^2 + 132(5.5) + 328320 which becomes:\r<BR>\n" );
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document.write( "y = 328683\r<BR>\n" );
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document.write( "your maximum revenue is at 328683\r<BR>\n" );
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document.write( "this occurs when x = 5.5\r<BR>\n" );
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document.write( "when x = 5.5, your price becomes 1.71 - .01 * 5.5 which becomes 1.71 - .055 which becomes 1.655.\r<BR>\n" );
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document.write( "your maximum revenue is when the price is equal to 1.655.\r<BR>\n" );
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document.write( "this can also be found by graphing the equation of y = -12x^2 + 132x + 328320 as shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/110904.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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