document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=788874>Question 788874</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A producer of synfuel from coal estimates that the cost C dollars per barrel for a production run of x thousand barrels is given by C = 9x^2 - 180x + 940. How many thousand barrels should be produced each run to keep the cost per barrel at a minimum? What is the minimum cost per barrel of synfuel?\r<BR>\n" );
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document.write( "Can you please help me out? Thanks so much in advance:)<BR>\n" );
document.write( "Can you also please show the steps it would really help me understand:) Thank-you  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #478700 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=478700\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+9x%5E2+-+180x+%2B+940 ALIGN=MIDDLE  ALT=\"C+=+9x%5E2+-+180x+%2B+940\"   DISABLED_style= \"max-width:90%; height:auto;\" > describes C as a quadratic function of x.<BR>\n" );
document.write( "The graph (with C as the y coordinate) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C40%2C-100%2C900%2C+9x%5E2+-+180x+%2B+940%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C40%2C-100%2C900%2C+9x%5E2+-+180x+%2B+940%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That curve is called a parabola, and the pointy end is called the vertex of the parabola.<BR>\n" );
document.write( "In this case it is a minimum value for C and it happens when x=10, and C=40.<BR>\n" );
document.write( "There is more than one way to find the coordinates (x and C) for that minimum point in the graph.<BR>\n" );
document.write( "You can apply formulas (memorized or copied, understood, or not).<BR>\n" );
document.write( "Another option is to \"complete the square\" transforming the equation into a form that will give away its secrets.<BR>\n" );
document.write( " <BR>\n" );
document.write( "WITH FORMULAS:<BR>\n" );
document.write( "A function of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3E0 ALIGN=MIDDLE  ALT=\"a%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is called a quadratic functiuon, and graphs as a parabola.<BR>\n" );
document.write( "Its axis of symmetry and vertex are at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F2a ALIGN=MIDDLE  ALT=\"x=-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "If you find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F2a ALIGN=MIDDLE  ALT=\"x=-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > value of the x-coordinate at the vertex, you can substitute it into the function to find the y-coordinate.<BR>\n" );
document.write( "Otherwise you can apply another, more complicated formula to find the y-coordinate at the vertex.<BR>\n" );
document.write( "The vertex will be a maximum of the function if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C0 ALIGN=MIDDLE  ALT=\"a%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and it will be a minimum if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3E0 ALIGN=MIDDLE  ALT=\"a%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "In this case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=9%3E0 ALIGN=MIDDLE  ALT=\"a=9%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so we have a minimum, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-180 ALIGN=MIDDLE  ALT=\"b=-180\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So, at the minimum <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-%28-180%29%2F%282%2A9%29=180%2F18=10 ALIGN=MIDDLE  ALT=\"x=-%28-180%29%2F%282%2A9%29=180%2F18=10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Plugging that value into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+9x%5E2+-+180x+%2B+940 ALIGN=MIDDLE  ALT=\"C+=+9x%5E2+-+180x+%2B+940\"   DISABLED_style= \"max-width:90%; height:auto;\" > we find the minimum value for C:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+9%2A10%5E2+-+180%2A10+%2B+940=9%2A100-1800%2B940=900-1800%2B940=1840-1800=highlight%2840%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "COMPLETING THE SQUARE:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+9x%5E2+-+180x+%2B+940 ALIGN=MIDDLE  ALT=\"C+=+9x%5E2+-+180x+%2B+940\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+9%2A%28x%5E2+-+20x%29+%2B+940 ALIGN=MIDDLE  ALT=\"C+=+9%2A%28x%5E2+-+20x%29+%2B+940\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+9%2A%28x%5E2+-+20x%2B100-100%29+%2B+940 ALIGN=MIDDLE  ALT=\"C+=+9%2A%28x%5E2+-+20x%2B100-100%29+%2B+940\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+9%2A%28x%5E2+-+20x%2B100%29-9%2A100+%2B+940 ALIGN=MIDDLE  ALT=\"C+=+9%2A%28x%5E2+-+20x%2B100%29-9%2A100+%2B+940\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+9%2A%28x%5E2+-+20x%2B100%29-900+%2B+940 ALIGN=MIDDLE  ALT=\"C+=+9%2A%28x%5E2+-+20x%2B100%29-900+%2B+940\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+9%2A%28x%5E2+-+20x%2B100%29%2B40 ALIGN=MIDDLE  ALT=\"C+=+9%2A%28x%5E2+-+20x%2B100%29%2B40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+9%2A%28x-10%29%5E2%2B40 ALIGN=MIDDLE  ALT=\"C+=+9%2A%28x-10%29%5E2%2B40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That last form of the equation tells you that there is a minimum, that the minimum will happens when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=10 ALIGN=MIDDLE  ALT=\"x=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=highlight%2840%29 ALIGN=MIDDLE  ALT=\"C=highlight%2840%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and that the curve is very pointy, getting 9 times taller that it is wide for x one unit away from its center, and rising very sharply to the left and right.  </SPAN>\n" );
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