document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=552093>Question 552093</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Among rectangles with a perimeter of 350 cm, what are the dimensions of the one having maximum area? Lesson is about the application of quadratic equations, and it requires a complete solution. 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 #360094 by </B><A HREF=/tutors/aboutme.mpl?userid=neatmath><B>neatmath(302)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=neatmath><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=neatmath><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=360094\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><font face=\"Tahoma\">I know the answer for this before I begin, but let's go through the steps!<br>\r<BR>\n" );
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document.write( "We are given a perimeter of 350 cm.<br>\r<BR>\n" );
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document.write( "Let L stand for length, and W stand for width.  Then,<br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2L%2B2W=350 ALIGN=MIDDLE  ALT=\"2L%2B2W=350\"   DISABLED_style= \"max-width:90%; height:auto;\" > given<br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%2BW=175 ALIGN=MIDDLE  ALT=\"L%2BW=175\"   DISABLED_style= \"max-width:90%; height:auto;\" > divided both sides by 2<br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=175-W ALIGN=MIDDLE  ALT=\"L=175-W\"   DISABLED_style= \"max-width:90%; height:auto;\" > subtracted W from both sides, solved in terms of L<br>\r<BR>\n" );
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document.write( "Now, we can use this information to \"maximize\" our area:<br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=L%2AW ALIGN=MIDDLE  ALT=\"A=L%2AW\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28175-W%29%2AW ALIGN=MIDDLE  ALT=\"A=%28175-W%29%2AW\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=175W-W%5E2 ALIGN=MIDDLE  ALT=\"A=175W-W%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "We can now set up a table and pick some values for W to solve this problem.<br>\r<BR>\n" );
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document.write( "Of course, W must be a value between 0 and 350 to satisfy our given info:\r<BR>\n" );
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document.write( "W=0, then A=0<BR>\n" );
document.write( "W=1, then A=174<BR>\n" );
document.write( "W=10, then A=1650<BR>\n" );
document.write( "W=80, then A=7600<BR>\n" );
document.write( "W=90, then A=7650<BR>\n" );
document.write( "W=100, then A=7500<BR>\n" );
document.write( "W=150, then A=3750<br>\r<BR>\n" );
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document.write( "Interesting, no?<br>\r<BR>\n" );
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document.write( "We can see that our Area is being maximized somewhere around 90.<br>\r<BR>\n" );
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document.write( "And in fact, if you do the math, the maximum area will occur where W=87.5<br>\r<BR>\n" );
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document.write( "If W=87.5, then A=7656.25<br>\r<BR>\n" );
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document.write( "If W=87.5, then L=87.5 as well.<br>\r<BR>\n" );
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document.write( "And you know what?  If the length and width of a rectangle are equal, then you have a square by definition.<br>\r<BR>\n" );
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document.write( "Given a set perimeter, the maximum area of a rectangle will ALWAYS, ALWAYS occur when the rectangle is in fact a square.<br>\r<BR>\n" );
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document.write( "That's how I knew the answer even before we started.<br>\r<BR>\n" );
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document.write( "I just needed to divide 350 by 4 (for each of the sides).<br>\r<BR>\n" );
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document.write( "And 350/4 is indeed equal to 87.5.<br>\r<BR>\n" );
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document.write( "Final answer:<br>\r<BR>\n" );
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document.write( "The dimensions of a rectangle with maximum area and a perimeter of 350 cm are length of 87.5 cm and width of 87.5 cm.<br>\r<BR>\n" );
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document.write( "This can also be easily done using Calculus, where we go through the same steps that we did above, but then use something called a derivative to more easily calculate the maximum area of the rectangle.<br>\r<BR>\n" );
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document.write( "But for now, this is the best algebraic way to solve this type of problem, in my opinion.<br>\r<BR>\n" );
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document.write( "I hope this helps!  :)<br>\r<BR>\n" );
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document.write( "Email Scott: neatmath@yahoo.com for help with specific problems,<br>\r<BR>\n" );
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document.write( "or to inquire about mathematics tutoring via email or other methods.<br>\r<BR>\n" );
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document.write( "Single problems would range from 50 cents to 5 dollars each, depending on their complexity.<br><br> </SPAN>\n" );
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