document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=191814>Question 191814</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An architect is designing an atrium for a hotel. The atrium is to be rectangular with a perimeter of 720ft of brass piping. What dimensions will maximize the area of the atrium? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #143939 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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=143939\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The perimeter of a rectangle is given by:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+2%28L%2BW%29 ALIGN=MIDDLE  ALT=\"P+=+2%28L%2BW%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and this is equal to 720. so...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28L%2BW%29+=+720 ALIGN=MIDDLE  ALT=\"2%28L%2BW%29+=+720\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 2 to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%2BW+=+360 ALIGN=MIDDLE  ALT=\"L%2BW+=+360\"   DISABLED_style= \"max-width:90%; height:auto;\" > which can be written as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L+=+360-W ALIGN=MIDDLE  ALT=\"L+=+360-W\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area of a rectangle is given by:<BR>\n" );
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;\" > Substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L+=+360-W ALIGN=MIDDLE  ALT=\"L+=+360-W\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+%28360-W%29%2AW ALIGN=MIDDLE  ALT=\"A+=+%28360-W%29%2AW\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+360W-W%5E2 ALIGN=MIDDLE  ALT=\"A+=+360W-W%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now you have a quadratic equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-W%5E2%2B360W+=+A ALIGN=MIDDLE  ALT=\"-W%5E2%2B360W+=+A\"   DISABLED_style= \"max-width:90%; height:auto;\" > When graphed, this will show a parabola opening downwards so there will be a maximum point on the curve and this occurs at the vertex of the parabola.  To find the value of W that corresponds to this point, use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W+=+%28-b%29%2F2a ALIGN=MIDDLE  ALT=\"W+=+%28-b%29%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > where a = -1 and b = 360.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W+=+%28-360%29%2F2%28-1%29 ALIGN=MIDDLE  ALT=\"W+=+%28-360%29%2F2%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W+=+180 ALIGN=MIDDLE  ALT=\"W+=+180\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the maximum area is obtained when W = 180ft. and L = 180ft.<BR>\n" );
document.write( "This should come as no surprise as it is well-known that the maximum area of a rectangle enclosed by a given perimeter is, in fact a square.\r<BR>\n" );
document.write( "\n" );
document.write( "Check:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+2%28L%2BW%29 ALIGN=MIDDLE  ALT=\"P+=+2%28L%2BW%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Substitute L = 180 and W = 180<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+2%28180%2B180%29 ALIGN=MIDDLE  ALT=\"P+=+2%28180%2B180%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+2%28360%29 ALIGN=MIDDLE  ALT=\"P+=+2%28360%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+720 ALIGN=MIDDLE  ALT=\"P+=+720\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...the given perimeter.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );