document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=159212>Question 159212</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangular patio is surrounded on three sides by a fence (the remaning side is up against the house). If the area of the patio is 38 meter square, and the total length of fence is 18 meters, what is the length and width of the patio. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #117357 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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=117357\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> it appears that the length can be any of 2 numbers and satisfy the equation.<BR>\n" );
document.write( "the equation is<BR>\n" );
document.write( "perimeter = l + 2*w = 18 (fence only covers 1 length)<BR>\n" );
document.write( "area = l*w = 38<BR>\n" );
document.write( "in the first equation, we take l = 18 - (2*w) and substitute for l in the second equation.<BR>\n" );
document.write( "the second equation becomes<BR>\n" );
document.write( "(18 - 2*w)*w = 38<BR>\n" );
document.write( "this expands to<BR>\n" );
document.write( "18*w - 2*w^2 = 38<BR>\n" );
document.write( "subtracting both sides of the equation by 18*w and adding 2*w to both sides of the equation and it becomes<BR>\n" );
document.write( "0 = 2*w^2 - 18*w + 38<BR>\n" );
document.write( "which is the same as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Aw%5E2+-+18%2Aw+%2B+38+=+0 ALIGN=MIDDLE  ALT=\"2%2Aw%5E2+-+18%2Aw+%2B+38+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the answer is not apparent right away so we'll use the quadratic equation to solve.<BR>\n" );
document.write( "quadratic equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w=%28%28-b%29%2Bsqrt%28b%5E2-4%2Aa%2Ac%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"w=%28%28-b%29%2Bsqrt%28b%5E2-4%2Aa%2Ac%29%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w=%28%28-b%29-sqrt%28b%5E2-4%2Aa%2Ac%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"w=%28%28-b%29-sqrt%28b%5E2-4%2Aa%2Ac%29%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "(-b) = +18<BR>\n" );
document.write( "a = 2<BR>\n" );
document.write( "c = 38<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28b%5E2-4%2Aa%2Ac%29+=+sqrt%2820%29 ALIGN=MIDDLE  ALT=\"sqrt%28b%5E2-4%2Aa%2Ac%29+=+sqrt%2820%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+=+%2818+%2B+sqrt%2820%29%29%2F4%29 ALIGN=MIDDLE  ALT=\"w+=+%2818+%2B+sqrt%2820%29%29%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 5.618033989<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%2818+-+sqrt%2820%29%29%2F4%29 ALIGN=MIDDLE  ALT=\"w+=+%2818+-+sqrt%2820%29%29%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 3.381966011<BR>\n" );
document.write( "solving for l, we get<BR>\n" );
document.write( "18 - 2*5.618033989 = 6.763932923<BR>\n" );
document.write( "18 - 2*3.381966011 = 11.23606798<BR>\n" );
document.write( "width can be 5.618033989 while length is 6.763932923, or<BR>\n" );
document.write( "width can be 3.381966011 while length is 11.23606798<BR>\n" );
document.write( "to prove,<BR>\n" );
document.write( "perimeter = l + 2*w = 18 (fence perimeter contained only 1*l)<BR>\n" );
document.write( "6.763932923 + 2*5.618033989 = 18.0000009 = 18 if the numbers weren't rounded because of the limitation of the calculator display.<BR>\n" );
document.write( "also<BR>\n" );
document.write( "11.23606798 + 2*3.381966011 = 18<BR>\n" );
document.write( "perimeter is ok.<BR>\n" );
document.write( "area = 2*l + 2*w<BR>\n" );
document.write( "6.763932923 * 5.618033989 = 37.99998431 = 38 if the numbers weren't rounded because of the limitation of the calculator display.<BR>\n" );
document.write( "11.23606798 * 3.381966011 = 38.00000001 = 38 if the numbers weren't rounded because of the limitation of the calculator display.<BR>\n" );
document.write( "to prove that for yourself, do the calculations using a calculator and store the results without truncating and then solve again using the numbers from inside the calculator rather than the numbers in the display.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );