document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=939772>Question 939772</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I figure out the smallest surface area of a cylinder when I only know what the volume is? the volume is 50,000cm cubed.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #572721 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=572721\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If you need to design a cylinder with a volume of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2250%2C000%22 ALIGN=MIDDLE  ALT=\"%2250%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E3 ALIGN=MIDDLE  ALT=\"cm%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "you could start with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" >= radius of the bases of the cylinder, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H ALIGN=MIDDLE  ALT=\"H\"   DISABLED_style= \"max-width:90%; height:auto;\" >= height of the cylinder,<BR>\n" );
document.write( "both in <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( "The volume in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E3 ALIGN=MIDDLE  ALT=\"cm%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2AR%5E2%2AH ALIGN=MIDDLE  ALT=\"pi%2AR%5E2%2AH\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If that volume is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2250%2C000%22 ALIGN=MIDDLE  ALT=\"%2250%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E3 ALIGN=MIDDLE  ALT=\"cm%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2AR%5E2%2AH=%2250%2C000%22 ALIGN=MIDDLE  ALT=\"pi%2AR%5E2%2AH=%2250%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H=%2250%2C000%22%2F%28pi%2AR%5E2%29 ALIGN=MIDDLE  ALT=\"H=%2250%2C000%22%2F%28pi%2AR%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Now, if you make it a right cylinder<BR>\n" );
document.write( "(one with its axis perpendicular to the bases,<BR>\n" );
document.write( "the total surface area is calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2pi%2AR%5E2%2B2pi%2AR%2AH ALIGN=MIDDLE  ALT=\"A=2pi%2AR%5E2%2B2pi%2AR%2AH\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H=%2250%2C000%22%2F%28pi%2AR%5E2%29 ALIGN=MIDDLE  ALT=\"H=%2250%2C000%22%2F%28pi%2AR%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the equation above, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2pi%2AR%5E2%2B2pi%2AR%2A%28%2250%2C000%22%2F%28pi%2AR%5E2%29%29 ALIGN=MIDDLE  ALT=\"A=2pi%2AR%5E2%2B2pi%2AR%2A%28%2250%2C000%22%2F%28pi%2AR%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2pi%2AR%5E2%2B2%2A%2250%2C000%22%2F%28pi%2AR%29%29 ALIGN=MIDDLE  ALT=\"A=2pi%2AR%5E2%2B2%2A%2250%2C000%22%2F%28pi%2AR%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2pi%2AR%5E2%2B%22100%2C000%22%2F%28pi%2AR%29%29 ALIGN=MIDDLE  ALT=\"A=2pi%2AR%5E2%2B%22100%2C000%22%2F%28pi%2AR%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area as a function of radius of the cylinder<BR>\n" );
document.write( "is very large for<BR>\n" );
document.write( "very narrow (and very tall) cylinders, and for<BR>\n" );
document.write( "very short and very wide) cylinders with the same area<BR>\n" );
document.write( "Its graph looks like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C40%2C-1000%2C9000%2C2pi%2Ax%5E2%2B100000%2F%28pi%2Ax%29%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C40%2C-1000%2C9000%2C2pi%2Ax%5E2%2B100000%2F%28pi%2Ax%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In the middle, there is a value for the radius that makes the surface are as small as possible.<BR>\n" );
document.write( "How would you find that value for the radius, and the minimum area?<BR>\n" );
document.write( "Maybe you are expected to use a graphing calculator.<BR>\n" );
document.write( "Maybe you are studying calculus.<BR>\n" );
document.write( "and are expected to calculate the derivative of the function<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2pi%2AR%5E2%2B%22100%2C000%22%2F%28pi%2AR%29%29 ALIGN=MIDDLE  ALT=\"A=2pi%2AR%5E2%2B%22100%2C000%22%2F%28pi%2AR%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and find the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > value where that derivative is zero ans <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > is minimum.<BR>\n" );
document.write( "The derivative is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2FdR=4pi%2AR-100000%2F%28pi%2AR%5E2%29 ALIGN=MIDDLE  ALT=\"dA%2FdR=4pi%2AR-100000%2F%28pi%2AR%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2FdR=%284pi%5E2%2AR%5E3-100000%29%2F%28pi%2AR%5E2%29 ALIGN=MIDDLE  ALT=\"dA%2FdR=%284pi%5E2%2AR%5E3-100000%29%2F%28pi%2AR%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and that is zero when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4pi%5E2%2AR%5E3=100000 ALIGN=MIDDLE  ALT=\"4pi%5E2%2AR%5E3=100000\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E3=100000%2F4pi%5E2 ALIGN=MIDDLE  ALT=\"R%5E3=100000%2F4pi%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R=root%283%2C100000%2F4pi%5E2%29 ALIGN=MIDDLE  ALT=\"R=root%283%2C100000%2F4pi%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R=13.63%28rounded%29 ALIGN=MIDDLE  ALT=\"R=13.63%28rounded%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the approximate value for the minimum area (in cubic cm)<BR>\n" );
document.write( "(when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > takes that value) is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=3502.6 ALIGN=MIDDLE  ALT=\"A=3502.6\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );