document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182496>Question 1182496</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Mang Lino is planning to construct a rectangular coop with the width 2 meter shorter that thrice the length. He wants to use at least 4 meters of chicken wire but at most 20 meters of the material. What are the possible integral dimensions of the coop? What is the inequality statement? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #812557 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</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=812557\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "W = width<BR>\n" );
document.write( "L = length<BR>\n" );
document.write( "which are some positive whole numbers. The units for each are in meters.\r<BR>\n" );
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document.write( "The width is 2 meters shorter than thrice the length, meaning<BR>\n" );
document.write( "W = 3L-2<BR>\n" );
document.write( "since 3L is thrice, or three times, the length. Then we subtract off 2 from that result to get the width W.\r<BR>\n" );
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document.write( "The perimeter of the rectangle with length L and width W is<BR>\n" );
document.write( "P = 2L+2W<BR>\n" );
document.write( "we add up two copies of L and W each to get the total distance around the rectangle. This is the amount of fencing needed for the rectangular coop.\r<BR>\n" );
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document.write( "He wants to use at least 4 meters but at most 20 meters<BR>\n" );
document.write( "This places boundaries on how low and how high P can get<BR>\n" );
document.write( "P = 4 is the smallest possible perimeter<BR>\n" );
document.write( "P = 20 is the largest possible perimeter<BR>\n" );
document.write( "That leads us to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+%3C=+P+%3C=+20 ALIGN=MIDDLE  ALT=\"4+%3C=+P+%3C=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's replace P with 2L+2W to get<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+%3C=+2L%2B2W+%3C=+20 ALIGN=MIDDLE  ALT=\"4+%3C=+2L%2B2W+%3C=+20\"   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=4+%3C=+2%28L%2BW%29+%3C=+20 ALIGN=MIDDLE  ALT=\"4+%3C=+2%28L%2BW%29+%3C=+20\"   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=2+%3C=+L%2BW+%3C=+10 ALIGN=MIDDLE  ALT=\"2+%3C=+L%2BW+%3C=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In the last step, I divided everything by 2\r<BR>\n" );
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document.write( "Now let's replace W with 3L-2 and isolate L like so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+%3C=+L%2BW+%3C=+10 ALIGN=MIDDLE  ALT=\"2+%3C=+L%2BW+%3C=+10\"   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=2+%3C=+L%2B3L-2+%3C=+10 ALIGN=MIDDLE  ALT=\"2+%3C=+L%2B3L-2+%3C=+10\"   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=2+%3C=+4L-2+%3C=+10 ALIGN=MIDDLE  ALT=\"2+%3C=+4L-2+%3C=+10\"   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=2%2B2+%3C=+4L-2%2B2+%3C=+10%2B2 ALIGN=MIDDLE  ALT=\"2%2B2+%3C=+4L-2%2B2+%3C=+10%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Adding 2 to all sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+%3C=+4L+%3C=+12 ALIGN=MIDDLE  ALT=\"4+%3C=+4L+%3C=+12\"   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=4%2F4+%3C=+4L%2F4+%3C=+12%2F4 ALIGN=MIDDLE  ALT=\"4%2F4+%3C=+4L%2F4+%3C=+12%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Dividing all sides by 4\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%3C=+L+%3C=+3 ALIGN=MIDDLE  ALT=\"1+%3C=+L+%3C=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This tells us that L = 1 is the smallest length possible while L = 3 is the largest length possible. L = 2 is also possible as it's right in between. These are the only integer lengths possible that meet the required conditions.\r<BR>\n" );
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document.write( "If L = 1, then<BR>\n" );
document.write( "W = 3L-2<BR>\n" );
document.write( "W = 3(1)-2<BR>\n" );
document.write( "W = 1<BR>\n" );
document.write( "So L = 1 leads to W = 1<BR>\n" );
document.write( "The perimeter would be P = 2L+2W = 2(1)+2(1) = 4<BR>\n" );
document.write( "So this is the case where he uses the least amount of wire possible (4 meters of it).\r<BR>\n" );
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document.write( "If L = 2, then,<BR>\n" );
document.write( "W = 3L-2<BR>\n" );
document.write( "W = 3(2)-2<BR>\n" );
document.write( "W = 4<BR>\n" );
document.write( "So L = 2 leads to W = 4<BR>\n" );
document.write( "The perimeter would be P = 2L+2W = 2(2)+2(4) = 12<BR>\n" );
document.write( "He is now using 12 meters of wire. This fits the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+%3C=+P+%3C=+20 ALIGN=MIDDLE  ALT=\"4+%3C=+P+%3C=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" > because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+%3C=+12+%3C=+20 ALIGN=MIDDLE  ALT=\"4+%3C=+12+%3C=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a true statement.\r<BR>\n" );
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document.write( "Lastly, if L = 3, then<BR>\n" );
document.write( "W = 3L-2<BR>\n" );
document.write( "W = 3(3)-2<BR>\n" );
document.write( "W = 7<BR>\n" );
document.write( "So L = 3 leads to W = 7<BR>\n" );
document.write( "The perimeter would be P = 2L+2W = 2(3)+2(7) = 20<BR>\n" );
document.write( "Telling us that he's using the most wire he's alloting or budgeting out. \r<BR>\n" );
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document.write( "To recap everything:<ul><li>The 1 m by 1 m coop needs 4 m of wire </li><li>The 2 m by 4 m coop needs 12 m of wire </li><li>The 3 m by 7 m coop needs 20 m of wire</li></ul>These are the only three possible cases if we want integral (ie integer) dimensions to the rectangle. <BR>\n" );
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