document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1033164>Question 1033164</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You have been asked to design a can shaped like a right circular cylinder with height h and radius r. Given that the can must hold exactly 410 cm3, what values of h and r will minimise the total surface area (including the top and bottom faces)? Give your answers correct to 2 decimal places as a list [in brackets] of the form: [ h, r ] <BR>\n" );
document.write( "for constants h (height), r (radius), in that order. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #647830 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=647830\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the volume of the cylinder with height <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > and radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > (both in cm) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2Ar%5E2%2Ah=410 ALIGN=MIDDLE  ALT=\"pi%2Ar%5E2%2Ah=410\"   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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2Ar%5E2%2Ah=410 ALIGN=MIDDLE  ALT=\"pi%2Ar%5E2%2Ah=410\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=410%2F%28pi%2Ar%5E2%29 ALIGN=MIDDLE  ALT=\"h=410%2F%28pi%2Ar%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The total surface area (including the top and bottom faces)of a right circular cylinder with height <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > and radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2pi%2Ar%2Ah%2B2pi%2Ar%5E2 ALIGN=MIDDLE  ALT=\"A=2pi%2Ar%2Ah%2B2pi%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "(We would measure <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > and radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > in cm, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, of course).<BR>\n" );
document.write( "Substituting the expression found for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   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%2A%28410%2F%28pi%2Ar%5E2%29%29%2B2pi%2Ar%5E2 ALIGN=MIDDLE  ALT=\"A=2pi%2Ar%2A%28410%2F%28pi%2Ar%5E2%29%29%2B2pi%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2%2A410%2Fr%2B2pi%2Ar%5E2 ALIGN=MIDDLE  ALT=\"A=2%2A410%2Fr%2B2pi%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2%2A%28410%2Bpi%2Ar%5E3%29%2Fr ALIGN=MIDDLE  ALT=\"A=2%2A%28410%2Bpi%2Ar%5E3%29%2Fr\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "We need to find the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > that yields the minimum for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There may be another way to find that value.<BR>\n" );
document.write( "Maybe you are expected to do it using a graphing calculator,<BR>\n" );
document.write( "or maybe you are expected to use calculus,<BR>\n" );
document.write( "and specifically derivatives.<BR>\n" );
document.write( "The result should be the same.<BR>\n" );
document.write( "Using calculus:<BR>\n" );
document.write( "A local minimum of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > happens only for a value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > that makes the derivative zero.<BR>\n" );
document.write( "The derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2%2A410%2Fr%2B2pi%2Ar%5E2 ALIGN=MIDDLE  ALT=\"A=2%2A410%2Fr%2B2pi%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdr=-2%2A410%2Fr%5E2%2B4pi%2Ar=%282%2Fr%5E2%29%28-410%2B2pi%2Ar%5E3%29 ALIGN=MIDDLE  ALT=\"dA%2Fdr=-2%2A410%2Fr%5E2%2B4pi%2Ar=%282%2Fr%5E2%29%28-410%2B2pi%2Ar%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Fr%5E2%29%28-410%2B2pi%2Ar%5E3%29=0 ALIGN=MIDDLE  ALT=\"%282%2Fr%5E2%29%28-410%2B2pi%2Ar%5E3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-410%2B2pi%2Ar%5E3=0 ALIGN=MIDDLE  ALT=\"-410%2B2pi%2Ar%5E3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E3=410%2F%282%2Api%29=205%2Fpi ALIGN=MIDDLE  ALT=\"r%5E3=410%2F%282%2Api%29=205%2Fpi\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=root%283%2C%28205%2Fpi%29%29=highlight%284.03%29 ALIGN=MIDDLE  ALT=\"r=root%283%2C%28205%2Fpi%29%29=highlight%284.03%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (correct to 2 decimal places).<BR>\n" );
document.write( "Then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=410%2F%28pi%2Ar%5E2%29=410%2F%28pi%2A%28205%2Fpi%29%5E%282%2F3%29%29=2%2A%28205%2Fpi%29%5E%282%2F3%29=highlight%288.05%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );