document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=90931>Question 90931</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0072549017/algebrahomeworkh>Beginning Al</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> I've racked my brain for hours & I still don't know where to begin on this problem: A rectangular garden is to be surrounded by a walkway of constant width. The garden's dimensions are 30 ft by 40 ft. the total area, garden plus walkway, is to be 1800 ft^2. What must be the width of the walkway to the nearest thousandth?\r<BR>\n" );
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document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #65999 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=65999\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If we draw a picture we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-5%2C5%2C-5%2C5%2C%0D%0Aline%28-5%2C5%2C5%2C5%29%2C%0D%0Aline%285%2C5%2C5%2C-5%29%2C%0D%0Aline%285%2C-5%2C-5%2C-5%29%2C%0D%0Aline%28-5%2C-5%2C-5%2C5%29%2C%0D%0A%0D%0Aarrow%28-3%2C3%2C-5%2C3%29%2C%0D%0Aarrow%28-5%2C3%2C-3%2C3%29%2C%0D%0Alocate%28-4%2C3%2Cx%29%2C%0D%0A%0D%0Aarrow%283%2C3%2C5%2C3%29%2C%0D%0Aarrow%285%2C3%2C3%2C3%29%2C%0D%0Alocate%284%2C3%2Cx%29%2C%0D%0A%0D%0A%0D%0Aarrow%283%2C3%2C3%2C5%29%2C%0D%0Aarrow%283%2C5%2C3%2C3%29%2C%0D%0Alocate%283%2C4%2Cx%29%2C%0D%0A%0D%0A%0D%0Aarrow%283%2C-3%2C3%2C-5%29%2C%0D%0Aarrow%283%2C-5%2C3%2C-3%29%2C%0D%0Alocate%283%2C-4%2Cx%29%2C%0D%0A%0D%0A%0D%0Alocate%280%2C3%2C30%29%2C%0D%0Alocate%283%2C0%2C40%29%2C%0D%0Arectangle%28-3%2C-3%2C3%2C3%29%0D%0A%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If we let x be the width of the path, notice there are 2 x-values per side. So we add 2x to 40 to get 40+2x. This is the total length of the side that contains 40 feet. \r<BR>\n" );
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document.write( "Also, this means we add 2x to 30 to get 30+2x. This is the total length of the side that contains 30 feet. \r<BR>\n" );
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document.write( "Start with the general area function:\r<BR>\n" );
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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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1800=xy ALIGN=MIDDLE  ALT=\"1800=xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1800 ALIGN=MIDDLE  ALT=\"A=1800\"   DISABLED_style= \"max-width:90%; height:auto;\" > (this is the total area)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1800=%2840%2B2x%29%2830%2B2x%29 ALIGN=MIDDLE  ALT=\"1800=%2840%2B2x%29%2830%2B2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=40%2B2x ALIGN=MIDDLE  ALT=\"x=40%2B2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=30%2B2x ALIGN=MIDDLE  ALT=\"y=30%2B2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1800=4x%5E2%2B140x%2B1200 ALIGN=MIDDLE  ALT=\"1800=4x%5E2%2B140x%2B1200\"   DISABLED_style= \"max-width:90%; height:auto;\" > Foil\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=4x%5E2%2B140x-600 ALIGN=MIDDLE  ALT=\"0=4x%5E2%2B140x-600\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 1800 from both sides\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for x:\r<BR>\n" );
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document.write( "Starting with the general quadratic\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the general solution using the quadratic equation is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ax%5E2%2B140%2Ax-600=0 ALIGN=MIDDLE  ALT=\"4%2Ax%5E2%2B140%2Ax-600=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=4 ALIGN=MIDDLE  ALT=\"a=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=140 ALIGN=MIDDLE  ALT=\"b=140\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-600 ALIGN=MIDDLE  ALT=\"c=-600\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-140+%2B-+sqrt%28+%28140%29%5E2-4%2A4%2A-600+%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"x+=+%28-140+%2B-+sqrt%28+%28140%29%5E2-4%2A4%2A-600+%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=4, b=140, and c=-600\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-140+%2B-+sqrt%28+19600-4%2A4%2A-600+%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"x+=+%28-140+%2B-+sqrt%28+19600-4%2A4%2A-600+%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 140 to get 19600  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-140+%2B-+sqrt%28+19600%2B9600+%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"x+=+%28-140+%2B-+sqrt%28+19600%2B9600+%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A-600%2A4 ALIGN=MIDDLE  ALT=\"-4%2A-600%2A4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9600 ALIGN=MIDDLE  ALT=\"9600\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-140+%2B-+sqrt%28+29200+%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"x+=+%28-140+%2B-+sqrt%28+29200+%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-140+%2B-+20%2Asqrt%2873%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"x+=+%28-140+%2B-+20%2Asqrt%2873%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root (note: If you need help with simplifying the square root, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-140+%2B-+20%2Asqrt%2873%29%29%2F8 ALIGN=MIDDLE  ALT=\"x+=+%28-140+%2B-+20%2Asqrt%2873%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 4 to get 8\r<BR>\n" );
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document.write( "So now the expression breaks down into two parts\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-140+%2B+20%2Asqrt%2873%29%29%2F8 ALIGN=MIDDLE  ALT=\"x+=+%28-140+%2B+20%2Asqrt%2873%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-140+-+20%2Asqrt%2873%29%29%2F8 ALIGN=MIDDLE  ALT=\"x+=+%28-140+-+20%2Asqrt%2873%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now break up the fraction\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-140%2F8%2B20%2Asqrt%2873%29%2F8 ALIGN=MIDDLE  ALT=\"x=-140%2F8%2B20%2Asqrt%2873%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-140%2F8-20%2Asqrt%2873%29%2F8 ALIGN=MIDDLE  ALT=\"x=-140%2F8-20%2Asqrt%2873%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-35+%2F+2%2B5%2Asqrt%2873%29%2F2 ALIGN=MIDDLE  ALT=\"x=-35+%2F+2%2B5%2Asqrt%2873%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-35+%2F+2-5%2Asqrt%2873%29%2F2 ALIGN=MIDDLE  ALT=\"x=-35+%2F+2-5%2Asqrt%2873%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So these expressions approximate to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3.86000936329383 ALIGN=MIDDLE  ALT=\"x=3.86000936329383\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-38.8600093632938 ALIGN=MIDDLE  ALT=\"x=-38.8600093632938\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our possible solutions are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3.86000936329383 ALIGN=MIDDLE  ALT=\"x=3.86000936329383\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-38.8600093632938 ALIGN=MIDDLE  ALT=\"x=-38.8600093632938\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since a negative length doesn't make sense, our only solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3.86000936329383 ALIGN=MIDDLE  ALT=\"x=3.86000936329383\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which is 3.860 to the nearest thousandth\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1800=%2840%2B2x%29%2830%2B2x%29 ALIGN=MIDDLE  ALT=\"1800=%2840%2B2x%29%2830%2B2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given area function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1800=%2840%2B2%283.860%29%29%2830%2B2%283.860%29%29 ALIGN=MIDDLE  ALT=\"1800=%2840%2B2%283.860%29%29%2830%2B2%283.860%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > plug in x=3.860\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1800=%2840%2B7.72%29%2830%2B7.72%29 ALIGN=MIDDLE  ALT=\"1800=%2840%2B7.72%29%2830%2B7.72%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1800=%2847.72%29%2837.72%29 ALIGN=MIDDLE  ALT=\"1800=%2847.72%29%2837.72%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1800=1799.9984 ALIGN=MIDDLE  ALT=\"1800=1799.9984\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since we rounded, this is as close as it gets. So our answer is verified.\r<BR>\n" );
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