document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=192419>Question 192419</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the dimensions of the cylinder of maximum volume that can be inscribed in a sphere with a radius of 8cm.\r<BR>\n" );
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document.write( "The 8cm is the sphere's radius. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #144508 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=144508\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Consider a diameter of the sphere perpendicular to the inscribed cylinder's axis.  From the center of the sphere, construct a radius that intersects the circle of intersection between the sphere and one base of the cylinder.  We need to derive a function for the volume of the cylinder in terms of the angle between the above described diameter and the constructed radius.  Let the measure of that angle be <i>x</i>.\r<BR>\n" );
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document.write( "Working this problem will be computationally simpler if we consider the sphere to have radius <i>r</i> for the time being.  Given that, the radius of the base of the cylinder is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ r_c(x) = r\cos{x}\">\r<BR>\n" );
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document.write( "And the area of the base of the cylinder is then:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A_b(x) = \pi r^2\cos^2{x}\">\r<BR>\n" );
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document.write( "Half the height of the cylinder is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ r\sin{x}\">\r<BR>\n" );
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document.write( "And then the height of the cylinder is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ h_c(x) = 2r\sin{x}\">\r<BR>\n" );
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document.write( "Now we can describe 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_c(x) = \pi \left(r^2\cos^2{x}\right)\left(2r\sin{x}\right)\">\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE V_c(0) = 0\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE V_c(\frac{\pi}{2}) = 0\"> and these are the limiting values for <i>x</i>, if we can find an extreme point for the function on the interval <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 0 < x < {\pi \over 2}\">, we can be assured that this point is a maximum.\r<BR>\n" );
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document.write( "Take the derivative of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE V_c\">:\r<BR>\n" );
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document.write( "First factor out the constants:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V_c(x) = 2\pi r^3 \left(\cos^2{x}\right)\left(\sin{x}\right)\">\r<BR>\n" );
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document.write( "Now define <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x) = \cos^2{x}\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE g(x) = \sin{x}\">\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 \ \ \ \ \ \ \ \ \ \ V_c(x) = 2\pi r^3f(x)g(x)\">\r<BR>\n" );
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document.write( "Now use the Product Rule:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V_c^{\,'}(x) = 2\pi r^3 f'(x)g(x) + f(x)g'(x)\">\r<BR>\n" );
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document.write( "Use the Chain Rule to derive <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f'(x)\">\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u = \cos{x}\">\r<BR>\n" );
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document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x)=y = u^2\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f'(x)=\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=(2\cos{x})(-\sin{x})=-2\cos{x}\sin{x} \">\r<BR>\n" );
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document.write( "Also:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE g'(x) = \cos{x}\">\r<BR>\n" );
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document.write( "So:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V_c^{\,'}(x) = 2\pi r^3 \left(\left(-2\cos{x}\sin{x}\right)\left( \sin{x} \right)  + \left(\cos^2{x}\right)\left(\cos{x}\right)\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V_c^{\,'}(x) = 2\pi r^3 \left(-2\cos{x}\sin^2{x}  + \cos^3{x}\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V_c^{\,'}(x) = 2\pi r^3\cos{x} \left(-2\sin^2{x}  + \cos^2{x}\right)\">\r<BR>\n" );
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document.write( "A local extreme (and here, a local maximum) is at the point where the first derivative is equal to zero, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\pi r^3\cos{x} \left(-2\sin^2{x}  + \cos^2{x}\right) = 0\">\r<BR>\n" );
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document.write( "Applying the zero product rule:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos{x} = 0\">\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -2\sin^2{x} + cos^2{x}=0\">\r<BR>\n" );
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document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos{x} = 0 \ \ \Rightarrow\ \ x = \frac{\pi}{2}\"> which is not in the feasible interval, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 0 < x < {\pi \over 2}\">, hence we exclude this value as an extraneous root.\r<BR>\n" );
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document.write( "To solve:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -2\sin^2{x} + cos^2{x}=0\">\r<BR>\n" );
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document.write( "Use\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos^2{x} = 1 - \sin^2{x}\">\r<BR>\n" );
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document.write( "So:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -2\sin^2{x} + 1 - \sin^2{x} = 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3 \sin^2{x} = 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sin^2{x} = \frac{1}{3}\">\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 \ \ \ \ \ \ \ \ \ \ \sin{x} = \frac{sqrt{3}}{3}\">\r<BR>\n" );
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document.write( "and, again using <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos^2{x} = 1 - \sin^2{x}\"> to determine:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos^2{x} = \frac{2}{3}\">\r<BR>\n" );
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document.write( "Substituting these values you can use the volume function to calculate the volume of the maximum inscribed cylinder for the general sphere of radius <i>r</i>:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V_c(x) = 2\pi r^3 \left(\frac{2}{3}\right)\left(\frac{sqrt{3}}{3}\right)=\frac{4\pi r^3\sqrt{3}}{9}\">\r<BR>\n" );
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document.write( "Notice that the calculation reduces to simply multiplying the volume of the sphere <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(\frac{4\pi r^3}{3}\right)\"> by a factor of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{sqrt{3}}{3}\">.\r<BR>\n" );
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document.write( "Of course, for your specific sphere of radius 8, you need to substitute for <i>r</i>.  I'll let you do the arithmetic.\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( "</font> </SPAN>\n" );
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