document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196071>Question 1196071</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The dimensions of a rectangular piece of paper ABCD are AB= 10 and BC = 9. It is folded so that corner D is matched with point F on edge BC. Given that length DE= 6, find EF,EC,FC, and the area EFC.\r<BR>\n" );
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document.write( "The lengths EF, EC, and FC are all functions of the length DE. The area of triangle EFC is also a function of DE. Using X to stand for DE, write formulas for these four functions,\r<BR>\n" );
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document.write( "Find the value of X that maximizes the area of triangle EFC.\r<BR>\n" );
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document.write( "Diagram link:\r<BR>\n" );
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document.write( "https://math.stackexchange.com/questions/2421004/areas-in-a-folded-sheet-of-paper<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 #828761 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=828761\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The referenced URL tells you how you can answer the questions; did you try that?<br><BR>\n" );
document.write( "EF is the same length as DE, so EF=x.<br><BR>\n" );
document.write( "DC is 10, and DE+EC=DC, so EC=10-x.<br><BR>\n" );
document.write( "Then use the Pythagorean Theorem to find an expression for FC:<br><BR>\n" );
document.write( "FC = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28x%5E2%29-%2810-x%29%5E2%29=sqrt%28x%5E2-%28x%5E2-20x%2B100%29%29=sqrt%2820x-100%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28x%5E2%29-%2810-x%29%5E2%29=sqrt%28x%5E2-%28x%5E2-20x%2B100%29%29=sqrt%2820x-100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And the area of EFC is one-half base times height:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%2810-x%29%28sqrt%2820x-100%29%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28%2810-x%29%28sqrt%2820x-100%29%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To find the value of x that maximizes the area of triangle EFC, by far the easiest thing to do is graph the area function on a graphing calculator and use the calculator's features to find the answer.<br><BR>\n" );
document.write( "Doing that shows the maximum area to be when x=20/3.<br><BR>\n" );
document.write( "Or you could use calculus, if you know the subject....<br><BR>\n" );
document.write( "(The factor of 1/2 in the area formula is not important for finding the maximum area, so I will ignore it in my work.)<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%2810-x%29%28%2820x-100%29%5E%281%2F2%29%29 ALIGN=MIDDLE  ALT=\"y=%2810-x%29%28%2820x-100%29%5E%281%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Use the product rule and chain rule to find the derivative.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx=%28-1%28%2820x-100%29%5E%281%2F2%29%29%29%2B%2810-x%29%28%281%2F2%29%28%2820x-100%29%5E%28-1%2F2%29%29%29%2820%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx=%28-%2820x-100%29%5E%281%2F2%29%29%2B%2810%2810-x%29%2F%2820x-100%29%5E%281%2F2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Find where the derivative is equal to zero.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-%2820x-100%29%5E%281%2F2%29%29%2B%2810%2810-x%29%2F%2820x-100%29%5E%281%2F2%29%29=0 ALIGN=MIDDLE  ALT=\"%28-%2820x-100%29%5E%281%2F2%29%29%2B%2810%2810-x%29%2F%2820x-100%29%5E%281%2F2%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%2820x-100%29%5E%281%2F2%29%29=%2810%2810-x%29%2F%2820x-100%29%5E%281%2F2%29%29 ALIGN=MIDDLE  ALT=\"%28%2820x-100%29%5E%281%2F2%29%29=%2810%2810-x%29%2F%2820x-100%29%5E%281%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2820x-100%29=10%2810-x%29 ALIGN=MIDDLE  ALT=\"%2820x-100%29=10%2810-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=20x-100=100-10x ALIGN=MIDDLE  ALT=\"20x-100=100-10x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30x=200 ALIGN=MIDDLE  ALT=\"30x=200\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=200%2F30=20%2F3 ALIGN=MIDDLE  ALT=\"x=200%2F30=20%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The calculus confirms the answer found using a graphing calculator.<br><BR>\n" );
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