document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=149714>Question 149714</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two square wire frames are to be constructed from a piece of wire 100 inches long. If the area enclosed by one frame is to be one-half the area enclosed by the other, find the dimensions of each frame. (Disregard the thickness of the wire.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #109839 by </B><A HREF=/tutors/aboutme.mpl?userid=nerdybill><B>nerdybill(7384)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nerdybill><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=nerdybill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=109839\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x = side of smaller square<BR>\n" );
document.write( "and y = side of larger square<BR>\n" );
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document.write( "x^2 is the area of smaller square<BR>\n" );
document.write( "y^2 is the area of larger square<BR>\n" );
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document.write( "4x is the perimeter of smaller square<BR>\n" );
document.write( "4y is the perimeter of larger square<BR>\n" );
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document.write( "Since we have two unknowns, we need two equations:<BR>\n" );
document.write( "From: \"Two square wire frames are to be constructed from a piece of wire 100 inches long.\" we get:<BR>\n" );
document.write( "4x + 4y = 100<BR>\n" );
document.write( "x + y = 25 (equation 1)<BR>\n" );
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document.write( "From: \"If the area enclosed by one frame is to be one-half the area enclosed by the other\" we get:<BR>\n" );
document.write( "x^2 = (1/2)y^2 (equation 2)<BR>\n" );
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document.write( "solving equation 1 for x:<BR>\n" );
document.write( "x + y = 25<BR>\n" );
document.write( "x = 25 - y<BR>\n" );
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document.write( "plug the above into equation 2 and solve for y:<BR>\n" );
document.write( "x^2 = (1/2)y^2<BR>\n" );
document.write( "(25-y)^2 = (1/2)y^2<BR>\n" );
document.write( "625 - 50y + y^2 = (1/2)y^2<BR>\n" );
document.write( "1250 - 100y + 2y^2 = y^2<BR>\n" );
document.write( "1250 - 100y + y^2 = 0<BR>\n" );
document.write( "y^2 - 100y + 1250 = 0<BR>\n" );
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document.write( "From the \"quadratic equation\" you'll get:<BR>\n" );
document.write( "y = 85.36 <BR>\n" );
document.write( "x = 14.64<BR>\n" );
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document.write( "Therefore the dimensions are:<BR>\n" );
document.write( "85.36 in by 85.36 in (larger square)<BR>\n" );
document.write( "14.64 in by 14.64 in (smaller square)<BR>\n" );
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document.write( "Details of the quadratic solution follows:<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ay%5E2%2Bby%2Bc=0 ALIGN=MIDDLE  ALT=\"ay%5E2%2Bby%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1y%5E2%2B-100y%2B1250+=+0 ALIGN=MIDDLE  ALT=\"1y%5E2%2B-100y%2B1250+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"y%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-100%29%5E2-4%2A1%2A1250=5000 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-100%29%5E2-4%2A1%2A1250=5000\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=5000 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--100%2B-sqrt%28+5000+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--100%2B-sqrt%28+5000+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D+=+%28-%28-100%29%2Bsqrt%28+5000+%29%29%2F2%5C1+=+85.3553390593274 ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+%28-%28-100%29%2Bsqrt%28+5000+%29%29%2F2%5C1+=+85.3553390593274\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D+=+%28-%28-100%29-sqrt%28+5000+%29%29%2F2%5C1+=+14.6446609406726 ALIGN=MIDDLE  ALT=\"y%5B2%5D+=+%28-%28-100%29-sqrt%28+5000+%29%29%2F2%5C1+=+14.6446609406726\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1y%5E2%2B-100y%2B1250 ALIGN=MIDDLE  ALT=\"1y%5E2%2B-100y%2B1250\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1y%5E2%2B-100y%2B1250+=+1%28y-85.3553390593274%29%2A%28y-14.6446609406726%29 ALIGN=MIDDLE  ALT=\"1y%5E2%2B-100y%2B1250+=+1%28y-85.3553390593274%29%2A%28y-14.6446609406726%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 85.3553390593274, 14.6446609406726.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-100%2Ax%2B1250+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-100%2Ax%2B1250+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );