document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=773353>Question 773353</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given a perimeter,P what is the largest area can you make for a quadrilateral? Express the area, A in terms of P. What kind of quadrilateral is this?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #471664 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=471664\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The question should be \"what is the largest area can you make for a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28rectangle%29 ALIGN=MIDDLE  ALT=\"highlight%28rectangle%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >?\"<BR>\n" );
document.write( "It is a much more difficult geometry/trigonometry problem if you have to prove that the largest area requires the quadrilateral to have 4 right angles.<BR>\n" );
document.write( "(If you had to prove that it has 4 right angles, ask again. If you do it as a thank you, I'll answer it myself).<BR>\n" );
document.write( " <BR>\n" );
document.write( "WARNING: I write a lot of explanations, extra information and comments (added for better understanding), and alternate ways to get to solutions. What follows is not a streamlined solution in the format of your teacher's preference.<BR>\n" );
document.write( " <BR>\n" );
document.write( "A rectangle has two parallel sides of length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, across from each other.<BR>\n" );
document.write( "The other two sides, are perpendicular to the first two sides, and have length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We do not care if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is greater, equal or less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >. It is a rectangle anyway.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C200%2C-3%2C33%2C-2%2C22%2C%0D%0Arectangle%280%2C0%2C30%2C20%29%2C%0D%0Alocate%2814%2C0%2Cx%29%2Clocate%2814%2C20%2Cx%29%2C%0D%0Alocate%280.5%2C11%2Cy%29%2Clocate%2830.5%2C11%2Cy%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The perimeter of the rectangle is calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=2%28x%2By%29 ALIGN=MIDDLE  ALT=\"P=2%28x%2By%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area of the rectangle is calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=xy ALIGN=MIDDLE  ALT=\"A=xy\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28P=2%28x%2By%29%2CA=xy%29 ALIGN=MIDDLE  ALT=\"system%28P=2%28x%2By%29%2CA=xy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28P%2F2=%28x%2By%29%2CA=xy%29 ALIGN=MIDDLE  ALT=\"system%28P%2F2=%28x%2By%29%2CA=xy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=P%2F2-x%2CA=xy%29 ALIGN=MIDDLE  ALT=\"system%28y=P%2F2-x%2CA=xy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=P%2F2-x%2CA=x%28P%2F2-x%29%29 ALIGN=MIDDLE  ALT=\"system%28y=P%2F2-x%2CA=x%28P%2F2-x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=P%2F2-x%2CA=%28P%2F2%29%2Ax-x%5E2%29 ALIGN=MIDDLE  ALT=\"system%28y=P%2F2-x%2CA=%28P%2F2%29%2Ax-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The quadratic function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=x%28P%2F2-x%29 ALIGN=MIDDLE  ALT=\"A=x%28P%2F2-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28P%2F2%29%2Ax-x%5E2%29 ALIGN=MIDDLE  ALT=\"A=%28P%2F2%29%2Ax-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has zeros at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=P%2F2%29 ALIGN=MIDDLE  ALT=\"x=P%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and<BR>\n" );
document.write( "a maximum for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=P%2F4 ALIGN=MIDDLE  ALT=\"x=P%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=P%2F4 ALIGN=MIDDLE  ALT=\"x=P%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > on, I see two ways to get to an expression for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > in terms of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=P%2F4%2CA=%28P%2F2%29%2Ax-x%5E2%29 ALIGN=MIDDLE  ALT=\"system%28x=P%2F4%2CA=%28P%2F2%29%2Ax-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28P%2F2%29%2A%28P%2F4%29-%28P%2F4%29%5E2%29 ALIGN=MIDDLE  ALT=\"A=%28P%2F2%29%2A%28P%2F4%29-%28P%2F4%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=P%2F8-P%2F16%29 ALIGN=MIDDLE  ALT=\"A=P%2F8-P%2F16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28A=P%2F16%29 ALIGN=MIDDLE  ALT=\"highlight%28A=P%2F16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "OR<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=P%2F4%2Cy=P%2F2-x%29 ALIGN=MIDDLE  ALT=\"system%28x=P%2F4%2Cy=P%2F2-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=P%2F4%2Cy=P%2F2-P%2F4%29 ALIGN=MIDDLE  ALT=\"system%28x=P%2F4%2Cy=P%2F2-P%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=P%2F4%2Cy=P%2F4%29 ALIGN=MIDDLE  ALT=\"system%28x=P%2F4%2Cy=P%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=P%2F4%2Cy=P%2F4%2CA=xy%29 ALIGN=MIDDLE  ALT=\"system%28x=P%2F4%2Cy=P%2F4%2CA=xy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28P%2F4%29%2A%28P%2F4%29 ALIGN=MIDDLE  ALT=\"A=%28P%2F4%29%2A%28P%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28A=P%2F16%29 ALIGN=MIDDLE  ALT=\"highlight%28A=P%2F16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The second option also tells you that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=P%2F4%2Cy=P%2F4%29 ALIGN=MIDDLE  ALT=\"system%28x=P%2F4%2Cy=P%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=y%29 ALIGN=MIDDLE  ALT=\"highlight%28x=y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That means that the rectangle with the greatest area is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28square%29 ALIGN=MIDDLE  ALT=\"highlight%28square%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "EXTRAS AND EXPLANATIONS:<BR>\n" );
document.write( "The graph of a quadratic function is a curve called a parabola.<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > graphed along the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > axis, the graph for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28P%2F2%29%2Ax-x%5E2%29 ALIGN=MIDDLE  ALT=\"A=%28P%2F2%29%2Ax-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > looks like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-0.1%2C0.9%2C-0.2%2C0.8%2C3.5x-5x%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-0.1%2C0.9%2C-0.2%2C0.8%2C3.5x-5x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "How do we know that the maximum (vertex of the parabola), and the axis of symmetry of the parabola are at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=P%2F4 ALIGN=MIDDLE  ALT=\"x=P%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >?<BR>\n" );
document.write( "For one thing, it is obvious that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28P%2F2%29%2Ax-x%5E2 ALIGN=MIDDLE  ALT=\"A=%28P%2F2%29%2Ax-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > has zeros at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=P%2F2%29 ALIGN=MIDDLE  ALT=\"x=P%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "and knowing that the function has a vertical axis of symmetry,<BR>\n" );
document.write( "we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the axis of symmetry (and the vertex/maximum that is part of that axis) has to be exactly in the middle,<BR>\n" );
document.write( "at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28P%2F2%2B0%29%2F2=P%2F4 ALIGN=MIDDLE  ALT=\"x=%28P%2F2%2B0%29%2F2=P%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So I would say it's obvious.<BR>\n" );
document.write( "However, if your teacher likes to see formulas, and you have to \"show your work\",<BR>\n" );
document.write( "the axis of symmetry and maximum for quadratic function of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "ias at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F2a ALIGN=MIDDLE  ALT=\"x=-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In the case of the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28P%2F2%29%2Ax-x%5E2 ALIGN=MIDDLE  ALT=\"A=%28P%2F2%29%2Ax-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=-x%5E2%2B%28P%2F2%29%2Ax%2B0 ALIGN=MIDDLE  ALT=\"A=-x%5E2%2B%28P%2F2%29%2Ax%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-1 ALIGN=MIDDLE  ALT=\"a=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=P%2F2 ALIGN=MIDDLE  ALT=\"b=P%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=0 ALIGN=MIDDLE  ALT=\"c=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so the axis of symmetry and maximum are at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28p%2F2%29%2F2 ALIGN=MIDDLE  ALT=\"x=%28p%2F2%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=P%2F4%29 ALIGN=MIDDLE  ALT=\"highlight%28x=P%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );