document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=173895>Question 173895</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangle has a perimeter of 140 Square feet. Maximize the area of the rectangle </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #128777 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=128777\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Perimeter:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=2L+%2B+2W ALIGN=MIDDLE  ALT=\"P=2L+%2B+2W\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Since the perimeter is 140 feet, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2L+%2B+2W+=+140 ALIGN=MIDDLE  ALT=\"2L+%2B+2W+=+140\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solving for L:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2L+=+140+-+2W ALIGN=MIDDLE  ALT=\"2L+=+140+-+2W\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L+=+70+-+W ALIGN=MIDDLE  ALT=\"L+=+70+-+W\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Area:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28LW%29+=+LW ALIGN=MIDDLE  ALT=\"A%28LW%29+=+LW\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but we know from the perimeter formula that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=70-W ALIGN=MIDDLE  ALT=\"L=70-W\"   DISABLED_style= \"max-width:90%; height:auto;\" > so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28W%29=%2870-W%29W=70W-W%5E2 ALIGN=MIDDLE  ALT=\"A%28W%29=%2870-W%29W=70W-W%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since A(W) is continuous and differentiable the first derivitive set to zero gives the value of the independent variable at a local extrema.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28dA%28W%29%29%2FdW=70-2W ALIGN=MIDDLE  ALT=\"%28dA%28W%29%29%2FdW=70-2W\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=70-2W=0 ALIGN=MIDDLE  ALT=\"70-2W=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W=35 ALIGN=MIDDLE  ALT=\"W=35\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since the first derivitive is also differentiable, the sign on the second derivitive at the extreme point will characterize the extreme as a maximum or minimum.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28d%5E2A%28W%29%29%2FdW%5E2=-2 ALIGN=MIDDLE  ALT=\"%28d%5E2A%28W%29%29%2FdW%5E2=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W ALIGN=MIDDLE  ALT=\"W\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the domain of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28W%29 ALIGN=MIDDLE  ALT=\"A%28W%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the extreme point is a maximum.\r<BR>\n" );
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document.write( "Therefore, a rectangle with perimeter 140 has a maximum area when the width is 35, which means that the length is also 35 and the rectangle is actually a square. </SPAN>\n" );
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