document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1001753>Question 1001753</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A company needs to make a cylindrical can that can hold precisely 0.7 liters of liquid. If the entire can is to be made out of the same material, find the dimensions of the can that will minimize the cost. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #618980 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=618980\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The volume of the can must be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=0.7L=700mL=700cm%5E3 ALIGN=MIDDLE  ALT=\"V=0.7L=700mL=700cm%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Let's define the measurements<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 base of the can, in cm<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 can, in cm<BR>\n" );
document.write( "As functions of those two variables, the Volume (in cubic centimeters) and the total surface Area of the can (in square centimeters) are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=pi%2AR%5E2%2Ah ALIGN=MIDDLE  ALT=\"V=pi%2AR%5E2%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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( "Since the volume of the can must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=700cm%5E3 ALIGN=MIDDLE  ALT=\"700cm%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28V=700%2CV=pi%2AR%5E2%2Ah%29 ALIGN=MIDDLE  ALT=\"system%28V=700%2CV=pi%2AR%5E2%2Ah%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2AR%5E2%2Ah=700 ALIGN=MIDDLE  ALT=\"pi%2AR%5E2%2Ah=700\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=700%2F%28pi%2AR%5E2%29 ALIGN=MIDDLE  ALT=\"h=700%2F%28pi%2AR%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting the expression fond 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;\" > in  <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;\" > , 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%28700%2F%28pi%2AR%5E2%29%29 ALIGN=MIDDLE  ALT=\"A=2pi%2AR%5E2%2B2pi%2AR%2A%28700%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%2B1400%2FR ALIGN=MIDDLE  ALT=\"A=2pi%2AR%5E2%2B1400%2FR\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "THE CALCULUS SOLUTION:<BR>\n" );
document.write( "The minimum for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2pi%2AR%5E2%2B1400%2FR ALIGN=MIDDLE  ALT=\"A=2pi%2AR%5E2%2B1400%2FR\"   DISABLED_style= \"max-width:90%; height:auto;\" > occurs 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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2FDR ALIGN=MIDDLE  ALT=\"dA%2FDR\"   DISABLED_style= \"max-width:90%; height:auto;\" > zero.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2FdR=4pi%2AR%2B1400%2A%28-1%2FR%5E2%29 ALIGN=MIDDLE  ALT=\"dA%2FdR=4pi%2AR%2B1400%2A%28-1%2FR%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2FdR=%284pi%2AR%5E3-1400%29%2FR%5E2%29 ALIGN=MIDDLE  ALT=\"dA%2FdR=%284pi%2AR%5E3-1400%29%2FR%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2FdR=0 ALIGN=MIDDLE  ALT=\"dA%2FdR=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4pi%2AR%5E3-1400=0 ALIGN=MIDDLE  ALT=\"4pi%2AR%5E3-1400=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E3=1400%2F4pi ALIGN=MIDDLE  ALT=\"R%5E3=1400%2F4pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E3=350%2Fpi ALIGN=MIDDLE  ALT=\"R%5E3=350%2Fpi\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R=root%283%2C350%2Fpi%29 ALIGN=MIDDLE  ALT=\"R=root%283%2C350%2Fpi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R=%28350%2Fpi%29%5E%28%221+%2F+3%22%29 ALIGN=MIDDLE  ALT=\"R=%28350%2Fpi%29%5E%28%221+%2F+3%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%3Croot%283%2C350%2Fpi%29 ALIGN=MIDDLE  ALT=\"R%3Croot%283%2C350%2Fpi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2FdR%3C0 ALIGN=MIDDLE  ALT=\"dA%2FdR%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > 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;\" > is decreasing.<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%3Eroot%283%2C350%2Fpi%29 ALIGN=MIDDLE  ALT=\"R%3Eroot%283%2C350%2Fpi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2FdR%3E0 ALIGN=MIDDLE  ALT=\"dA%2FdR%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > 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;\" > is increasing.<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R=root%283%2C350%2Fpi%29 ALIGN=MIDDLE  ALT=\"R=root%283%2C350%2Fpi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2FdR=0 ALIGN=MIDDLE  ALT=\"dA%2FdR=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > 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;\" > is minimum.<BR>\n" );
document.write( "So, the dimensions of the can that will minimize the cost are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28R=root%283%2C350%2Fpi%29%29 ALIGN=MIDDLE  ALT=\"highlight%28R=root%283%2C350%2Fpi%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28cm=about4.8cm%29 ALIGN=MIDDLE  ALT=\"highlight%28cm=about4.8cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=700%2F%28pi%2A%28350%2Fpi%29%5E%28%222+%2F+3%22%29%29 ALIGN=MIDDLE  ALT=\"h=700%2F%28pi%2A%28350%2Fpi%29%5E%28%222+%2F+3%22%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28h=2%2A%28350%2Fpi%29%5E%281%2F3%29%29 ALIGN=MIDDLE  ALT=\"highlight%28h=2%2A%28350%2Fpi%29%5E%281%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28cm=about9.6cm%29 ALIGN=MIDDLE  ALT=\"highlight%28cm=about9.6cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );