document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=309449>Question 309449</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The volume of a cylindrical can is 500 cubed cm. the material used to make the top and bottom costs .o12 cent/squared cm, the material used for the sides costs .01 cent/ squared cm, and the seam joining the top and bottom to the sides costs .015 cent/cm. what size can would cost the least to produce?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #221325 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=221325\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The can is a cylinder of radius R and length L.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=pi%2AR%5E2%2AL=500 ALIGN=MIDDLE  ALT=\"V=pi%2AR%5E2%2AL=500\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The total cost is the cost of the top and bottom plus the cost of the sidewall. The costs are based on total area.<BR>\n" );
document.write( "The top and bottom are both circles of radius R.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A=pi%2AR%5E2 ALIGN=MIDDLE  ALT=\"+A=pi%2AR%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The contribution to the total cost is,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Ct%2BCb=2%2Api%2AR%5E2%2A0.012%0D%0ATHe+sidewall+is+a+rectangle+of+length+%7B%7B%7B2piR ALIGN=MIDDLE  ALT=\"+Ct%2BCb=2%2Api%2AR%5E2%2A0.012%0D%0ATHe+sidewall+is+a+rectangle+of+length+%7B%7B%7B2piR\"   DISABLED_style= \"max-width:90%; height:auto;\" > and width <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L ALIGN=MIDDLE  ALT=\"L\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A=2%2Api%2ARL ALIGN=MIDDLE  ALT=\"+A=2%2Api%2ARL\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Its contribution to the total cost is,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Cs=2%2Api%2ARL%2A0.01 ALIGN=MIDDLE  ALT=\"+Cs=2%2Api%2ARL%2A0.01\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The final contributor to cost is the seam which is length L.<BR>\n" );
document.write( "Its contribution is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Cx=L%2A0.015 ALIGN=MIDDLE  ALT=\"+Cx=L%2A0.015\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The total cost equation is then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Ctot=Ct%2BCb%2BCs%2BCx=0.024%2Api%2AR%5E2%2B0.02%2Api%2ARL%2B0.015L+ ALIGN=MIDDLE  ALT=\"+Ctot=Ct%2BCb%2BCs%2BCx=0.024%2Api%2AR%5E2%2B0.02%2Api%2ARL%2B0.015L+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the volume equation you can get L as a function of R.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2AR%5E2%2AL=500 ALIGN=MIDDLE  ALT=\"pi%2AR%5E2%2AL=500\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E2=500%2F%28pi%2AL%29 ALIGN=MIDDLE  ALT=\"R%5E2=500%2F%28pi%2AL%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now substitute into the cost equation,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Ctot=0.024%2Api%2AR%5E2%2B0.02%2Api%2ARL%2B0.015L+ ALIGN=MIDDLE  ALT=\"+Ctot=0.024%2Api%2AR%5E2%2B0.02%2Api%2ARL%2B0.015L+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Ctot=0.024%2Api%2A%28500%2F%28pi%2AL%29%29%2B0.02%2Api%2Asqrt%28500%2F%28pi%2AL%29%29L%2B0.015L+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Ctot=12%2FL%2B0.7927%2Asqrt%28L%29%2B0.015L ALIGN=MIDDLE  ALT=\"+Ctot=12%2FL%2B0.7927%2Asqrt%28L%29%2B0.015L\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now you have total cost as the function of one variable.<BR>\n" );
document.write( "To find the minimum cost, take the derivative and set it equal to zero.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dC%2Fdt=-12%2FL%5E2%2B%281%2F2%29%280.7927%29%2Fsqrt%28L%29%2B0.015=0 ALIGN=MIDDLE  ALT=\"dC%2Fdt=-12%2FL%5E2%2B%281%2F2%29%280.7927%29%2Fsqrt%28L%29%2B0.015=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I solved this numerically in EXCEL and got <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28L=9.02%29 ALIGN=MIDDLE  ALT=\"highlight_green%28L=9.02%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From that, then use the volume equation to find R.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E2=500%2F%28pi%2AL%29 ALIGN=MIDDLE  ALT=\"R%5E2=500%2F%28pi%2AL%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E2=500%2F%28pi%2A9.02%29 ALIGN=MIDDLE  ALT=\"R%5E2=500%2F%28pi%2A9.02%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28R=4.20%29 ALIGN=MIDDLE  ALT=\"highlight_green%28R=4.20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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