document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209264>Question 1209264</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How can I prove a square is the largest rectanglar area possible? I'm really stuck. Thanks for any help </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847937 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847937\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Your question seems a bit vague. <BR>\n" );
document.write( "My interpretation is that you're given some amount of fencing (a fixed constant perimeter P) and your teacher wants you to prove that the largest rectangle area happens when it's a square of side length 0.25P<BR>\n" );
document.write( "Recall that any square is a rectangle but not vice versa.\r<BR>\n" );
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document.write( "The formula for the perimeter of rectangle is<BR>\n" );
document.write( "P = 2*L + 2*W<BR>\n" );
document.write( "That can be rearranged to <BR>\n" );
document.write( "W = 0.5P - L<BR>\n" );
document.write( "The algebra shouldn't be too tricky.<BR>\n" );
document.write( "Let me know if you have questions about this portion.<BR>\n" );
document.write( "Replace 0.5 with 1/2 if you want.\r<BR>\n" );
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document.write( "Next I'll use x in place of L.<BR>\n" );
document.write( "x = length<BR>\n" );
document.write( "0.5P - x = width\r<BR>\n" );
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document.write( "area = length*width<BR>\n" );
document.write( "A = x*(0.5P - x)<BR>\n" );
document.write( "A = 0.5Px - x^2<BR>\n" );
document.write( "A = -x^2 + 0.5Px<BR>\n" );
document.write( "y = -x^2 + 0.5Px<BR>\n" );
document.write( "This graphs an upside down parabola because the leading coefficient is negative. You can use a graphing tool like <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> or <a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> (there are many others to choose from). <BR>\n" );
document.write( "Try graphing an example function like y = -x^2+4x or y = -x^2+10x to see what I mean.\r<BR>\n" );
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document.write( "Anyways let's return back to y = -x^2 + 0.5Px<BR>\n" );
document.write( "The highest point of this parabola is the vertex. <BR>\n" );
document.write( "It's where the rectangular area maxes out.\r<BR>\n" );
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document.write( "The roots occur when y = 0<BR>\n" );
document.write( "x*(0.5P - x) = 0<BR>\n" );
document.write( "x = 0 or 0.5P - x = 0<BR>\n" );
document.write( "x = 0 or x = 0.5P\r<BR>\n" );
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document.write( "If x = 0 or x = 0.5P, then the area is 0.<BR>\n" );
document.write( "The midpoint of the roots is the x coordinate of the vertex due to the parabola's symmetry.<BR>\n" );
document.write( "The midpoint of 0 and 0.5P is 0.25P<BR>\n" );
document.write( "Therefore the x coordinate of the vertex is x = 0.25P<BR>\n" );
document.write( "length = x = 0.25P<BR>\n" );
document.write( "width = 0.5P - x = 0.5P - 0.25P = 0.25P\r<BR>\n" );
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document.write( "Both length and width are 0.25P<BR>\n" );
document.write( "We have a square with side length 0.25P<BR>\n" );
document.write( "This concludes the proof. \r<BR>\n" );
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document.write( "We have proven that given a fixed perimeter (P), the max rectangular area occurs when the length and width are 0.25P (i.e. when we have a square)\r<BR>\n" );
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document.write( "If you're confused about the wording \"fixed perimeter\" it just means you're given some amount of fencing and cannot change the number. <BR>\n" );
document.write( "For instance if you're given P = 200 feet of fencing then 0.25*P = 0.25*200 = 50 feet is the side length of the square and the area is 50^2 = 2500 square feet. This is the largest rectangle area possible for this amount of fencing. <BR>\n" );
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