document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=683794>Question 683794</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello,the problem I'm trying to solve is,\r<BR>\n" );
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document.write( "\"A plane with a capacity of 120 passengers is to be chartered for an excursion. The price of one ticket is to be $60 if 100 people or fewer buy tickets. The airline will reduce the price of every ticket by 50 cents for each ticket sold over 100. How many additional passengers over 100 will bring about maximum income?\"\r<BR>\n" );
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document.write( "I know the first part of the equation will probably be (120 + 100x), but I'm not to sure about the rest of the equation which I need in order to graph the parabola. Help would be greatly appriciated. Thanks. \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #423745 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=423745\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the income will go up <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=linearly ALIGN=MIDDLE  ALT=\"linearly\"   DISABLED_style= \"max-width:90%; height:auto;\" > for each passenger until the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100th ALIGN=MIDDLE  ALT=\"100th\"   DISABLED_style= \"max-width:90%; height:auto;\" > one, where the income would be $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60+%2A+100+=+6000 ALIGN=MIDDLE  ALT=\"60+%2A+100+=+6000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "after that, evidently, the next passenger would pay only $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=59.50 ALIGN=MIDDLE  ALT=\"59.50\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100 ALIGN=MIDDLE  ALT=\"100\"   DISABLED_style= \"max-width:90%; height:auto;\" > other passengers would also pay $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.50 ALIGN=MIDDLE  ALT=\"0.50\"   DISABLED_style= \"max-width:90%; height:auto;\" > less\r<BR>\n" );
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document.write( "let's put that in equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Income+=+%2860+-%28n-100%29%2F2%29%2An ALIGN=MIDDLE  ALT=\"Income+=+%2860+-%28n-100%29%2F2%29%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "where \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" >\" is the number of passenger\r<BR>\n" );
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document.write( "note that this equation applies only for \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" >\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=greater ALIGN=MIDDLE  ALT=\"greater\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=equal ALIGN=MIDDLE  ALT=\"equal\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100 ALIGN=MIDDLE  ALT=\"100\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n-100%29%2F2 ALIGN=MIDDLE  ALT=\"%28n-100%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the bit that takes away <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50 ALIGN=MIDDLE  ALT=\"50\"   DISABLED_style= \"max-width:90%; height:auto;\" > cents for each passenger above the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100th ALIGN=MIDDLE  ALT=\"100th\"   DISABLED_style= \"max-width:90%; height:auto;\" >; and the \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" >\" is the number of passenger\r<BR>\n" );
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document.write( "A bit of algebra turns the income equation to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=income=110n+-n%5E2%2F2+ ALIGN=MIDDLE  ALT=\"income=110n+-n%5E2%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now, we want to maximize the income. Calculus to the rescue, as the local maximum implies that the derivative of the equation equals zero.\r<BR>\n" );
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document.write( "The derivative of the equation is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=110+-+n ALIGN=MIDDLE  ALT=\"110+-+n\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If we put this equal to zero, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > has to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=110 ALIGN=MIDDLE  ALT=\"110\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "That is thus the number of passengers that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=maximizes ALIGN=MIDDLE  ALT=\"maximizes\"   DISABLED_style= \"max-width:90%; height:auto;\" > income.\r<BR>\n" );
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document.write( "Plugging <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=110 ALIGN=MIDDLE  ALT=\"n=110\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the income equation means that said income would be <BR>\n" );
document.write( "$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6050 ALIGN=MIDDLE  ALT=\"6050\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "And we can easily verify that this is indeed the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=maximum ALIGN=MIDDLE  ALT=\"maximum\"   DISABLED_style= \"max-width:90%; height:auto;\" >, by seeing what the income would be for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=109 ALIGN=MIDDLE  ALT=\"109\"   DISABLED_style= \"max-width:90%; height:auto;\" > passengers (it would be $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6049.50 ALIGN=MIDDLE  ALT=\"6049.50\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=111 ALIGN=MIDDLE  ALT=\"111\"   DISABLED_style= \"max-width:90%; height:auto;\" > passengers ($<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6049.50 ALIGN=MIDDLE  ALT=\"6049.50\"   DISABLED_style= \"max-width:90%; height:auto;\" > also).\r<BR>\n" );
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