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 #619233 by </B><A HREF=/tutors/aboutme.mpl?userid=vleith><B>vleith(2983)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=vleith><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=619233\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the ticket price be given by the formula (170-x)  where x is the number of pennies in price reduction<BR>\n" );
document.write( "You are told the ridership formula is (192,000 + 1200x)  (for each reduction of a penny you get 1200 more riders)\r<BR>\n" );
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document.write( "Total revenue is the product of those two formulas<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=revenue+=+%28192000%2B1200x%29%28170-x%29 ALIGN=MIDDLE  ALT=\"revenue+=+%28192000%2B1200x%29%28170-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=revenue+=+32640000%2B12000x-1200x%5E2 ALIGN=MIDDLE  ALT=\"revenue+=+32640000%2B12000x-1200x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1200x%5E2%2B12000x%2B32640000+=+0 ALIGN=MIDDLE  ALT=\"-1200x%5E2%2B12000x%2B32640000+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%2812000%29%5E2-4%2A-1200%2A32640000=156816000000 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2812000%29%5E2-4%2A-1200%2A32640000=156816000000\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=156816000000 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-12000%2B-sqrt%28+156816000000+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-12000%2B-sqrt%28+156816000000+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%2812000%29%2Bsqrt%28+156816000000+%29%29%2F2%5C-1200+=+-160 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%2812000%29%2Bsqrt%28+156816000000+%29%29%2F2%5C-1200+=+-160\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%2812000%29-sqrt%28+156816000000+%29%29%2F2%5C-1200+=+170 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%2812000%29-sqrt%28+156816000000+%29%29%2F2%5C-1200+=+170\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1200x%5E2%2B12000x%2B32640000 ALIGN=MIDDLE  ALT=\"-1200x%5E2%2B12000x%2B32640000\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1200x%5E2%2B12000x%2B32640000+=+-1200%28x--160%29%2A%28x-170%29 ALIGN=MIDDLE  ALT=\"-1200x%5E2%2B12000x%2B32640000+=+-1200%28x--160%29%2A%28x-170%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -160, 170.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1200%2Ax%5E2%2B12000%2Ax%2B32640000+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1200%2Ax%5E2%2B12000%2Ax%2B32640000+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "The maximum revenue will be found at the midpoint of the two zeros. (170-(-160))/2 = 115\r<BR>\n" );
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document.write( "Does that match what you would get? </SPAN>\n" );
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