document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1122448>Question 1122448</A>: <I><SPAN STYLE=\"word-break: break-all;\"> problem solving using quadratic functions: <BR>\n" );
document.write( "Assume the perimeter of a rectangle is P and that the area is to be maximized. Let x represent the width of the rectangle. Using quadratic functions, show that the rectangle must be a square.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #738564 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=738564\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <H2>A rectangle with a given perimeter which has the maximal area is a square</H2>\r<BR>\n" );
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document.write( "There are many rectangles with a given perimeter.<BR>\n" );
document.write( "For example, if the perimeter is &nbsp;200 feet, &nbsp;you can consider rectangles &nbsp;20 by 80 &nbsp;feet, &nbsp;30 by 70 &nbsp;feet, &nbsp;40 by 60 feet, &nbsp;50 by 50 &nbsp;feet and many others.\r<BR>\n" );
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document.write( "The amazing fact is:  &nbsp;among rectangles with a given perimeter, &nbsp;a square has the maximal area.\r<BR>\n" );
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document.write( "<H3>Theorem</H3>Among rectangles with a given perimeter, &nbsp;a square has the maximal area.\r<BR>\n" );
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document.write( "<B>Proof</B>\r<BR>\n" );
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document.write( "Let P be the given perimeter of a rectangle, and let L be the length and W be the width of the rectangle.\r\n" );
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document.write( "Then we have \r\n" );
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document.write( "2L + 2W = P,\r\n" );
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document.write( "so we can express W as\r\n" );
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document.write( "W = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28P-2L%29%2F2 ALIGN=MIDDLE  ALT=\"%28P-2L%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r\n" );
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document.write( "The area of the rectangle is \r\n" );
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document.write( "L*W = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28L%2A%28P-2L%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28L%2A%28P-2L%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-L%5E2+%2B+%281%2F2%29%2APL ALIGN=MIDDLE  ALT=\"-L%5E2+%2B+%281%2F2%29%2APL\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r\n" );
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document.write( "<BLOCKQUOTE><TABLE BORDER=2>\r\n" );
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document.write( "  <TD>  Now let me remind you that, if you have a quadratic function f(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the general form,   \r\n" );
document.write( "    then it reaches the maximum/minimum at x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a ALIGN=MIDDLE  ALT=\"-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.</TD>\r\n" );
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document.write( "      See the lessons\r\n" );
document.write( "          - <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>HOW TO complete the square to find the minimum/maximum of a quadratic function</A>\r\n" );
document.write( "          - <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-How-to-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>Briefly on finding the minimum/maximum of a quadratic function</A>\r\n" );
document.write( "      in this site.\r\n" );
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document.write( "For our situation,  L = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28P%2F2%29%29%2F%28%282%29%29 ALIGN=MIDDLE  ALT=\"%28%28P%2F2%29%29%2F%28%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2F4 ALIGN=MIDDLE  ALT=\"P%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r\n" );
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document.write( "Then W = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28P-2L%29%2F2 ALIGN=MIDDLE  ALT=\"%28P-2L%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28P-2%2A%28P%2F4%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28P-2%2A%28P%2F4%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2F4 ALIGN=MIDDLE  ALT=\"P%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,\r\n" );
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document.write( "and the rectangle is the square with L = W =  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2F4 ALIGN=MIDDLE  ALT=\"P%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r\n" );
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document.write( "The statement is proved.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Let me repeat the statement one more time: \r<BR>\n" );
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document.write( "<BLOCKQUOTE><TABLE  BORDER=2>\n" );
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document.write( "  <TD >&nbsp;&nbsp;&nbsp;&nbsp;Among rectangles with a given perimeter &nbsp;P, &nbsp;a square with the side length &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2F4 ALIGN=MIDDLE  ALT=\"P%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp;&nbsp; has the maximal area, &nbsp;which is equal to &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28P%2F4%29%5E2 ALIGN=MIDDLE  ALT=\"%28P%2F4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.&nbsp;&nbsp;&nbsp;&nbsp;<BR>\n" );
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document.write( "</TABLE></BLOCKQUOTE>\r<BR>\n" );
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document.write( "<H3>Example 1</H3>A rectangle has a perimeter of &nbsp;100 meters. &nbsp;What are the dimensions of the sides if the area is a maximum? \r<BR>\n" );
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document.write( "<B>Solution</B>\r<BR>\n" );
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document.write( "According to the proved statement above, &nbsp;the rectangle is a square with the side length of &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%2F4 ALIGN=MIDDLE  ALT=\"100%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 25 meters.\r<BR>\n" );
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document.write( "<H3>Problem 1</H3>A piece of wire &nbsp;20 inches long is to be cut into a rectangle. &nbsp;How long should each side of the rectangle be to maximize the area of the rectangle? \r<BR>\n" );
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document.write( "<B>Solution</B>\r<BR>\n" );
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document.write( "According to the proved statement above, &nbsp;the rectangle should be a square with the side length of &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%2F4 ALIGN=MIDDLE  ALT=\"20%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 5 inches.\r<BR>\n" );
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document.write( "<H3>Problem 2</H3>A farmer has &nbsp;280 feet of fence to enclose a rectangular garden. &nbsp;What dimensions for the garden give the maximum area? \r<BR>\n" );
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document.write( "<B>Solution</B>\r<BR>\n" );
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document.write( "According to the proved statement above, &nbsp;the rectangular garden should be a square with the side length of &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=280%2F4 ALIGN=MIDDLE  ALT=\"280%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 70 feet.\r<BR>\n" );
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document.write( "What you see in this post, is the lesson\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/A-rectangle-with-the-given-perimeter-which-has-the-maximal-area-is-a-square.lesson>A rectangle with a given perimeter which has the maximal area is a square</A>\r<BR>\n" );
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document.write( "from this site.\r<BR>\n" );
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document.write( "Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.\r<BR>\n" );
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document.write( "The referred lessons are the part of this textbook under the topic \"<U>Finding minimum/maximum of quadratic functions</U>\". \r<BR>\n" );
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document.write( "Save the link to this online textbook together with its description\r<BR>\n" );
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document.write( "Free of charge online textbook in ALGEBRA-I<BR>\n" );
document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r<BR>\n" );
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document.write( "to your archive and use it when it is needed.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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