document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=975337>Question 975337</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2. A cylindrical can is to hold 16&#960;cm3 of mango juice. The cost per square meter of constructing the top and bottom is twice the cost per square meter of constructing the cardboard side. What dimensions minimize the total cost of construction? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #597105 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=597105\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "The volume of a cylinder is given by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V\ =\ \pi r^2h\">, so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  r^2h\ =\ 16\">\r<BR>\n" );
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document.write( "hence\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  r^2\ =\ \frac{16}{h}\">\r<BR>\n" );
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document.write( "The area of either the top or bottom of the can is given by\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \pi r^2\">,\r<BR>\n" );
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document.write( "but there are two ends, so the area of the ends is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  2\pi r^2\">\r<BR>\n" );
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document.write( "The area of the side of the can is the circumference of the end times the height of the can, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  2\pi h\">\r<BR>\n" );
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document.write( "The total surface area of the can is then:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  A\ =\ 2\pi r^2\ +\ 2\pi h\">\r<BR>\n" );
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document.write( "But if we substitute <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r^2\ =\ \frac{16}{h}\"> we can create a single variable function for the total surface area:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  A(h)\ =\ \frac{32pi}{h}\ +\ 2\pi h\">\r<BR>\n" );
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document.write( "Using the fact that the ends cost twice as much per unit area than the side, we can now write a function for the cost in terms of the height:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  C(h)\ =\ \frac{64pi}{h}\ +\ 2\pi h\">\r<BR>\n" );
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document.write( "Then take the first derivative:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dC}{dh}\ =\ 2\pi\ -\ \frac{64\pi}{h^2}\">\r<BR>\n" );
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document.write( "Set the first derivative equal to zero and solve for the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large h\"> that yields a local extrema for the function:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\pi h^2\ =\ 64\pi\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ h_{ext}\ =\ 4\sqrt{2}\">\r<BR>\n" );
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document.write( "Now take the 2nd derivative:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2C}{dh^2}\ =\ \frac{128\pi}{h^3}\">\r<BR>\n" );
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document.write( "Which is positive for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large h\ >\ 0\">, hence <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large h_{ext}\ =\ h_{min}\">\r<BR>\n" );
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document.write( "Now that we have a height value that minimizes the cost function, we can calculate the resulting radius value using the volume relationship.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  r^2\ =\ \frac{16}{4\sqrt{2}}\ =\ 2\sqrt{2}\ =\ \sqrt{8}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  r\ =\ \sqrt{\sqrt{8}}\ =\ \sqrt[4]{8}\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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