document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1030182>Question 1030182</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A right circular cone is inscribed inside a larger right circular cone with a volume of 150 cm3.  The axes of the cones coincide and the vertex of the inner cones touches the center of the base of the outer cone.  Find the ratio of the heights of the cones that maximizes the volume of the inner cone. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #645120 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=645120\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> By not giving enough data to find the radius of height, or height to radius ratio of the larger cone, the problem's wording hints that the answer does not depend on the height to radius ratio of the larger cone.<BR>\n" );
document.write( "I do not believe you need to know the volume of the larger cone, either.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If there is a way to solve the problem without invoking calculus,<BR>\n" );
document.write( "or if there is an easier or more elegant way to reach the answer,<BR>\n" );
document.write( "let me know.<BR>\n" );
document.write( "The way I saw to the solution is shown below.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Here is a cross-section of the cones, sliced along their axes.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-8.5%2C8.5%2C-1%2C16%2C%0D%0Agreen%28triangle%28-4%2C5%2C4%2C5%2C0%2C0%29%29%2C%0D%0Agreen%28rectangle%280%2C5%2C0.5%2C4.5%29%29%2C%0D%0Atriangle%286%2C0%2C0%2C0%2C0%2C15%29%2C%0D%0Atriangle%28-6%2C0%2C0%2C0%2C0%2C15%29%2C%0D%0Aarrow%287%2C7.5%2C7%2C0%29%2Carrow%287%2C7.5%2C7%2C15%29%2C%0D%0Alocate%287.1%2C8%2CH%29%2C%0D%0Agreen%28arrow%28-7.5%2C0%2C-7.5%2C5%29%29%2Cgreen%28arrow%28-7.5%2C5%2C-7.5%2C0%29%29%2C%0D%0Alocate%28-7.4%2C3%2Cgreen%28h%29%29%2Clocate%28-7.4%2C11%2CH-green%28h%29%29%2C%0D%0Aarrow%28-7.5%2C10%2C-7.5%2C15%29%2Carrow%28-7.5%2C10%2C-7.5%2C5%29%2C%0D%0Alocate%28-3%2C0.8%2CR%29%2Clocate%283%2C0.8%2CR%29%2C%0D%0Alocate%28-2%2C5%2Cgreen%28r%29%29%2Clocate%282%2C5%2Cgreen%28r%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The cones heights are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H ALIGN=MIDDLE  ALT=\"H\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > ; their radii are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The portion of the large cone above the base of the small cone is<BR>\n" );
document.write( "a cone similar to the large cone (same shape, but scaled down).<BR>\n" );
document.write( "Therefore, their corresponding length measurements are proportional:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2FR=%28H-h%29%2FH ALIGN=MIDDLE  ALT=\"r%2FR=%28H-h%29%2FH\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2FR=1-h%2FH ALIGN=MIDDLE  ALT=\"r%2FR=1-h%2FH\"   DISABLED_style= \"max-width:90%; height:auto;\" > <-- <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=R%281-h%2FH%29 ALIGN=MIDDLE  ALT=\"r=R%281-h%2FH%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The volume of the small cone is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v=pi%2Ar%5E2h%2F3=%28pi%2F3%29r%5E2h ALIGN=MIDDLE  ALT=\"v=pi%2Ar%5E2h%2F3=%28pi%2F3%29r%5E2h\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%281-h%2FH%29 ALIGN=MIDDLE  ALT=\"R%281-h%2FH%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   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=v=%28pi%2F3%29%28R%281-h%2FH%29%29%5E2h ALIGN=MIDDLE  ALT=\"v=%28pi%2F3%29%28R%281-h%2FH%29%29%5E2h\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v=%28pi%2F3%29R%5E2%281-h%2FH%29%5E2h ALIGN=MIDDLE  ALT=\"v=%28pi%2F3%29R%5E2%281-h%2FH%29%5E2h\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v=%28pi%2F3%29R%5E2%281-2h%2FH%2Bh%5E2%2FH%5E2%29h ALIGN=MIDDLE  ALT=\"v=%28pi%2F3%29R%5E2%281-2h%2FH%2Bh%5E2%2FH%5E2%29h\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v=%28pi%2F3%29R%5E2%28h-2h%5E2%2FH%2Bh%5E3%2FH%5E2%29 ALIGN=MIDDLE  ALT=\"v=%28pi%2F3%29R%5E2%28h-2h%5E2%2FH%2Bh%5E3%2FH%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is a function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" >, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H ALIGN=MIDDLE  ALT=\"H\"   DISABLED_style= \"max-width:90%; height:auto;\" > being constants.<BR>\n" );
document.write( "The derivative is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dv%2Fdr=%28pi%2F3%29R%5E2%281%2B4h%2FH%2B3h%5E2%2FH%5E2%29 ALIGN=MIDDLE  ALT=\"dv%2Fdr=%28pi%2F3%29R%5E2%281%2B4h%2FH%2B3h%5E2%2FH%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The maxima and minima of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" > will happen when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dv%2Fdr=0 ALIGN=MIDDLE  ALT=\"dv%2Fdr=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that will happen when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B4h%2FH%2B3h%5E2%2FH%5E2=0 ALIGN=MIDDLE  ALT=\"1%2B4h%2FH%2B3h%5E2%2FH%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If you change the name of the variables to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=h%2FH ALIGN=MIDDLE  ALT=\"x=h%2FH\"   DISABLED_style= \"max-width:90%; height:auto;\" > , you can re-write the equation as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-4x%2B1=0 ALIGN=MIDDLE  ALT=\"3x%5E2-4x%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and you would recognize it as a quadratic equation,<BR>\n" );
document.write( "with solutions <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F3 ALIGN=MIDDLE  ALT=\"x=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >(meaning <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28h%2FH=1%2F3%29 ALIGN=MIDDLE  ALT=\"highlight%28h%2FH=1%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > )<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (meaning <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2FH=1 ALIGN=MIDDLE  ALT=\"h%2FH=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).<BR>\n" );
document.write( "The polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B4h%2FH%2B3h%5E2%2FH%5E2=0 ALIGN=MIDDLE  ALT=\"1%2B4h%2FH%2B3h%5E2%2FH%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dv%2Fdr ALIGN=MIDDLE  ALT=\"dv%2Fdr\"   DISABLED_style= \"max-width:90%; height:auto;\" > changes sign at each of its two zeros,<BR>\n" );
document.write( "so one must be a maximum and the other a minimum.  <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2FH=1 ALIGN=MIDDLE  ALT=\"h%2FH=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > makes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=0 ALIGN=MIDDLE  ALT=\"r=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v=0 ALIGN=MIDDLE  ALT=\"v=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2FH=1 ALIGN=MIDDLE  ALT=\"h%2FH=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > the small cone volume is minimum.<BR>\n" );
document.write( "The maximum volume for the small cone happens when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28h%2FH=1%2F3%29 ALIGN=MIDDLE  ALT=\"highlight%28h%2FH=1%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );