document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198424>Question 1198424</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Cynthia Besch wants to buy a rug for a room that is 22 ft wide and 30 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy  560 square feet of carpeting. What dimensions should the rug​ have? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832010 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=832010\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>20 ft by 28 ft</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=========================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Explanation:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = width of the uniform strip (i.e. the gap between the rug edge and the wall)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Diagram<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=math_tutor2020&illustration_name=Screenshot_146.png\"><BR>\n" );
document.write( "The larger outer rectangle is 22 by 30<BR>\n" );
document.write( "The smaller inner rectangle inside is (22-2x) by (30-2x)<BR>\n" );
document.write( "For each inner dimension, we're subtracting two copies of x from the outer dimension.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The smaller rectangle area is 560 sq ft which is the amount of carpeting she can afford.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "length*width = area<BR>\n" );
document.write( "(22-2x)*(30-2x) = 560<BR>\n" );
document.write( "22*(30-2x) - 2x*(30-2x) = 560<BR>\n" );
document.write( "660-44x - 60x + 4x^2 = 560<BR>\n" );
document.write( "4x^2 - 104x + 660 = 560<BR>\n" );
document.write( "4x^2 - 104x + 660-560 = 0<BR>\n" );
document.write( "4x^2 - 104x + 100 = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Compare this to ax^2+bx+c = 0<BR>\n" );
document.write( "and we see that:<BR>\n" );
document.write( "a = 4<BR>\n" );
document.write( "b = -104<BR>\n" );
document.write( "c = 100\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plug those values into the quadratic formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-104%29%2B-sqrt%28%28-104%29%5E2-4%284%29%28100%29%29%29%2F%282%284%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-104%29%2B-sqrt%28%28-104%29%5E2-4%284%29%28100%29%29%29%2F%282%284%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28104%2B-sqrt%2810816-1600%29%29%2F%288%29 ALIGN=MIDDLE  ALT=\"x+=+%28104%2B-sqrt%2810816-1600%29%29%2F%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28104%2B-sqrt%289216%29%29%2F%288%29 ALIGN=MIDDLE  ALT=\"x+=+%28104%2B-sqrt%289216%29%29%2F%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28104%2B-++96%29%2F%288%29 ALIGN=MIDDLE  ALT=\"x+=+%28104%2B-++96%29%2F%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28104%2B96%29%2F%288%29 ALIGN=MIDDLE  ALT=\"x+=+%28104%2B96%29%2F%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28104-96%29%2F%288%29 ALIGN=MIDDLE  ALT=\"x+=+%28104-96%29%2F%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28200%29%2F%288%29 ALIGN=MIDDLE  ALT=\"x+=+%28200%29%2F%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%288%29%2F%288%29 ALIGN=MIDDLE  ALT=\"x+=+%288%29%2F%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+25 ALIGN=MIDDLE  ALT=\"x+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1 ALIGN=MIDDLE  ALT=\"x+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Those are the potential gap widths from the edge of the carpet to the wall.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We need to check each to see if they would be valid.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If x = 25, then<BR>\n" );
document.write( "22-2x = 22-2*25 = -28<BR>\n" );
document.write( "which isn't valid. A negative length doesn't make sense.<BR>\n" );
document.write( "Therefore we ignore x = 25. It's not a valid solution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If x = 1, then<BR>\n" );
document.write( "22-2x = 22-2*1 = 20<BR>\n" );
document.write( "30-2x = 30-2*1 = 28<BR>\n" );
document.write( "Both results are positive, so x = 1 is valid.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The carpet has dimensions of 20 ft by 28 ft<BR>\n" );
document.write( "Check: 20*28 = 560 sq ft<BR>\n" );
document.write( "The answers are confirmed.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );