document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=114296>Question 114296</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Draw and label a diagram for problem 1.Write answers as complete sentences\r<BR>\n" );
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document.write( "1.  A rain gutter is made from sheets of aluminum that are 20 inches wide by turning up the edges to form right angles.  A)  Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow.  B)  What is the maximum cross-sectional area?    </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #83177 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=83177\">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=drawing%28600%2C400%2C-1%2C12%2C-1%2C8%2C%0D%0Aline%280%2C0%2C0%2C5%29%2C%0D%0Aline%280%2C0%2C10%2C0%29%2C%0D%0Aline%2810%2C0%2C10%2C5%29%2C%0D%0Alocate%2810.5%2C2.5%2C+x%29%2C%0D%0Alocate%284.9%2C-.5%2C20-2x%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The cross-sectional area is given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bcs%5D=f%28x%29=x%2820-2x%29=-2x%5E2%2B20x ALIGN=MIDDLE  ALT=\"A%5Bcs%5D=f%28x%29=x%2820-2x%29=-2x%5E2%2B20x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "A critical point for f(x) is the value of x that satisfies <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%28x%29%2Fdx=0 ALIGN=MIDDLE  ALT=\"df%28x%29%2Fdx=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%28x%29%2Fdx=-4x%2B20 ALIGN=MIDDLE  ALT=\"df%28x%29%2Fdx=-4x%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4x%2B20=0 ALIGN=MIDDLE  ALT=\"-4x%2B20=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hence x = 5 is a critical point.  A critical point is a maximum if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2f%28x%29%2Fdx%5E2+%3C0 ALIGN=MIDDLE  ALT=\"d%5E2f%28x%29%2Fdx%5E2+%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2f%28x%29%2Fdx%5E2+=-4%3C0 ALIGN=MIDDLE  ALT=\"d%5E2f%28x%29%2Fdx%5E2+=-4%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, therefore the critical point at x = 5 is a maximum.\r<BR>\n" );
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document.write( "Therefore, the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow is 5 inches.\r<BR>\n" );
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document.write( "The maximum cross-sectional area is the value of the area function at 5.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bmax%5D=f%285%29=-2%285%29%5E2%2B20%285%29=-50%2B100=50 ALIGN=MIDDLE  ALT=\"A%5Bmax%5D=f%285%29=-2%285%29%5E2%2B20%285%29=-50%2B100=50\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Therefore the maximum cross-sectional area is 50 square inches.\r<BR>\n" );
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document.write( "Hope that helps.<BR>\n" );
document.write( "John </SPAN>\n" );
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