document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204467>Question 1204467</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ms.Ikelyn, <BR>\n" );
document.write( "I asked a question here: https://www.algebra.com/tutors/students/your-answer.mpl?question=1204433\r<BR>\n" );
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document.write( "To which you responded (with a helpful!) lesson about a TWIN problem. However, I am confused and need help. What is that \"shift\" term? And can you break it down simpler?\r<BR>\n" );
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document.write( "Thank you Ma'am </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840745 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=840745\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Let us assume these bottles are circles with radius 5cm for simplicity.\r<BR>\n" );
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document.write( "The densest possible packing is hexagonal packing for circles, as shown below:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF=\"https://imgbb.com/\"><IMG STYLE=\"max-width:80%;\" src=\"https://i.ibb.co/jfTW3mT/HJh78.png\" alt=\"HJh78\" border=\"0\"></a>\r<BR>\n" );
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document.write( "For the first layer, we find that the maximum amount of spheres we can fit on the first layer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000%2F%285%2A2%29=100+ ALIGN=MIDDLE  ALT=\"1000%2F%285%2A2%29=100+\"   DISABLED_style= \"max-width:90%; height:auto;\" >circles. \r<BR>\n" );
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document.write( "Since the second layer is in between the circles, one can only fit <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=99 ALIGN=MIDDLE  ALT=\"99\"   DISABLED_style= \"max-width:90%; height:auto;\" > circles. The third layer can fit another <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100+ ALIGN=MIDDLE  ALT=\"100+\"   DISABLED_style= \"max-width:90%; height:auto;\" >circles and so on.\r<BR>\n" );
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document.write( "Then, we can build an equilateral triangle and show that every stack increases the width of the area filled by \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2Asqrt%283%29=5sqrt%283%29 ALIGN=MIDDLE  ALT=\"r%2Asqrt%283%29=5sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " Given the first layer has a thickness (remember, circle represents bottles) of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10cm ALIGN=MIDDLE  ALT=\"10cm\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the amount of layers that can be fitted is the floor of \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281000-10%29%2F%285sqrt%283%29%29%29%2B1=115.3153532995459 ALIGN=MIDDLE  ALT=\"%28%281000-10%29%2F%285sqrt%283%29%29%29%2B1=115.3153532995459\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=115 ALIGN=MIDDLE  ALT=\"115\"   DISABLED_style= \"max-width:90%; height:auto;\" > layers. \r<BR>\n" );
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document.write( "since first two layers have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28100%2B99%29+ ALIGN=MIDDLE  ALT=\"%28100%2B99%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >circles, and there is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=115%2F2=57.5 ALIGN=MIDDLE  ALT=\"115%2F2=57.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=57 ALIGN=MIDDLE  ALT=\"57\"   DISABLED_style= \"max-width:90%; height:auto;\" >*first two layers have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28100%2B99%29 ALIGN=MIDDLE  ALT=\"%28100%2B99%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=.5=1%2F2 ALIGN=MIDDLE  ALT=\".5=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > means we take one more first layer which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100 ALIGN=MIDDLE  ALT=\"100\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This can be formed into \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=57%28100%2B99%29%2B100+=11443 ALIGN=MIDDLE  ALT=\"57%28100%2B99%29%2B100+=11443\"   DISABLED_style= \"max-width:90%; height:auto;\" > circles, in your case bottles.\r<BR>\n" );
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document.write( "answer: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11443 ALIGN=MIDDLE  ALT=\"11443\"   DISABLED_style= \"max-width:90%; height:auto;\" > bottles\r<BR>\n" );
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document.write( " </SPAN>\n" );
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