document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1061893>Question 1061893</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If one were to cut five circles, each 2 cm in diameter, from a rectangular piece of paper 6 cm long, at<BR>\n" );
document.write( "least how many cm’s wide would the piece of paper have to be? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #676710 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=676710\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here are 5 circles (with their centers) packed as tightly as I can pack them: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C450%2C-1.5%2C2.9%2C-3.3%2C3.3%2C%0D%0Agreen%28rectangle%28-1%2C-3%2Csqrt%283%29%2C3%29%29%2C%0D%0Agreen%28rectangle%280%2C-3%2C1%2Bsqrt%283%29%2C3%29%29%2C%0D%0Acircle%280%2C0%2C0.05%29%2Ccircle%280%2C0%2C1%29%2C%0D%0Acircle%280%2C2%2C0.05%29%2Ccircle%280%2C2%2C1%29%2C%0D%0Acircle%280%2C-2%2C0.05%29%2Ccircle%280%2C-2%2C1%29%2C%0D%0Acircle%28sqrt%283%29%2C1%2C0.05%29%2Ccircle%28sqrt%283%29%2C1%2C1%29%2C%0D%0Acircle%28sqrt%283%29%2C-1%2C0.05%29%2Ccircle%28sqrt%283%29%2C-1%2C1%29%2C%0D%0Ared%28triangle%280%2C0%2C0%2C-2%2Csqrt%283%29%2C-1%29%29%2C%0D%0Ared%28triangle%280%2C-1%2C0%2C0%2Csqrt%283%29%2C-1%29%29%2C%0D%0Ared%28rectangle%280%2C-1%2C0.2%2C-0.8%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If each circle has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=diameter=2cm ALIGN=MIDDLE  ALT=\"diameter=2cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=radius=1cm ALIGN=MIDDLE  ALT=\"radius=1cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the green rectangles have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=length=6cm ALIGN=MIDDLE  ALT=\"length=6cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For the rectangle to the left of all centers,<BR>\n" );
document.write( "and for the one to the right of all centers,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=width=1cm ALIGN=MIDDLE  ALT=\"width=1cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "As for the (middle) green rectangle going through all the circle centers,<BR>\n" );
document.write( "looking at the red equilateral and right triangles<BR>\n" );
document.write( "we can calculate its width.<BR>\n" );
document.write( "For the right triangles,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=hypotenuse=2cm ALIGN=MIDDLE  ALT=\"hypotenuse=2cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=short ALIGN=MIDDLE  ALT=\"short\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=leg=1cm ALIGN=MIDDLE  ALT=\"leg=1cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
document.write( "\n" );
document.write( "So, according to the Pythagorean theorem,<BR>\n" );
document.write( "the long leg measures<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%282cm%29%5E2%2B%281cm%29%5E2%29=sqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%28%282cm%29%5E2%2B%281cm%29%5E2%29=sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm ALIGN=MIDDLE  ALT=\"cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So the overall width encased in green (in cm) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B1%2Bsqrt%283%29=highlight%282%2Bsqrt%283%29=about3.732%29 ALIGN=MIDDLE  ALT=\"1%2B1%2Bsqrt%283%29=highlight%282%2Bsqrt%283%29=about3.732%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So I would say the width of the piece of paper has to be at least<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%283.74cm%29 ALIGN=MIDDLE  ALT=\"highlight%283.74cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or more realistically <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%283.8cm%29 ALIGN=MIDDLE  ALT=\"highlight%283.8cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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