document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=538428>Question 538428</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangular sheet of perimeter 40 cm and dimensions x cm by y cm is to be rolled into a cylinder.  What values of x and y given the largest cylinder volume? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #353178 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=353178\">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( "Since the assignment of the variables to the dimensions of the rectangle is purely arbitrary, we can assume that the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> dimension of the rectangle becomes the height dimension of the cylider.  Our first task is to develop a function V, representing the volume of the cylinder, as a function of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">.\r<BR>\n" );
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document.write( "First, we know that the perimeter of the rectangle is 40, so we can write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2x\ +\ 2y\ =\ 40\">\r<BR>\n" );
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document.write( "From which we can derive\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 20\ -\ x\">\r<BR>\n" );
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document.write( "The volume of a cylinder is found by multiplying the height by the area of the circular base.  Since the rectangle will be rolled into a cylinder, the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> dimension of the rectangle will become the circumference of the circular base, hence the radius of the circular base is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ r\ =\ \frac{x}{2\pi}\">\r<BR>\n" );
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document.write( "Using the formula for the volume of a cylinder, i.e. <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V\ =\ \pi{r}^2h\">, and substituting the facts derived so far, we get a function of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> that represents the volume of the cylinder:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V(x)\ =\ \pi\left(\frac{x}{2\pi}\right)^2\left(20\ -\ x\right)\">\r<BR>\n" );
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document.write( "A little Algebra music, Mr. Spear...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V(x)\ =\ \frac{20x^2\ -\ x^3}{4\pi}\">\r<BR>\n" );
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document.write( "Find the extrema:  Set the first derivitive equal to zero.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dV}{dx}\ =\ \frac{40x\ -\ 3x^2}{4\pi}\">\r<BR>\n" );
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document.write( "Said function having zeros at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 0\">, which we can discard because it is perforce a minimum, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{40}{3}\">.\r<BR>\n" );
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document.write( "Evaluate the second derivitive at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{40}{3}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2V}{dx^2}\ =\ \frac{40\ -\ 6x}{4\pi}\">\r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{40\ -\ 6\left(\frac{40}{3}\right)}{4\pi}\ <\ 0\">\r<BR>\n" );
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document.write( "Therefore the extremum at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \frac{40}{3}\"> is a maximum.\r<BR>\n" );
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document.write( "Plug <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \frac{40}{3}\"> back into <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ 20\ -\ x\"> to determine the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> dimension.\r<BR>\n" );
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document.write( "<i><b>Super-deluxe Double-plus Extra Credit</b></i>\r<BR>\n" );
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document.write( "Derive the ratio between the circumference of the circular base and the height of a cylinder of maximum volume for a given cylinder height.\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<BR>\n" );
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