document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1075217>Question 1075217</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sphere with radius 3 is inscribed in a conical frustum of slant height 10. (The sphere is tangent to both bases and the side of the frustum.) Find the volume of the frustum. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #689974 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=689974\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's look at a section of that frustum/cone/sphere<BR>\n" );
document.write( "cut center of the sphere, and the center of the base of the cone<BR>\n" );
document.write( "(and perpendicular to those bases):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28700%2C300%2C-10.5%2C10.5%2C-1%2C8%2C%0D%0Agreen%28triangle%280%2C0%2C0%2C6.75%2C9%2C0%29%29%2C%0D%0Agreen%28triangle%280%2C0%2C0%2C3%2C9%2C0%29%29%2C%0D%0Agreen%28rectangle%280%2C0%2C0.2%2C0.2%29%29%2C%0D%0Aline%28-9%2C0%2C9%2C0%29%2Cline%28-9%2C0%2C0%2C6.75%29%2C%0D%0Aline%289%2C0%2C0%2C6.75%29%2Ccircle%280%2C3%2C0.1%29%2C%0D%0Ared%28circle%280%2C3%2C3%29%29%2Ctriangle%289%2C0%2C1%2C0%2C1%2C6%29%2C%0D%0Arectangle%281%2C0%2C1.2%2C0.2%29%2Cline%28-1%2C6%2C1%2C6%29%2C%0D%0Alocate%281.05%2C3.5%2C6%29%2Clocate%283.5%2C3.9%2C10%29%2C%0D%0Alocate%284%2C0.5%2C8%29%2Cgreen%28arc%289%2C0%2C6%2C6%2C180%2C216.87%29%29%2C%0D%0Alocate%285.6%2C1.1%2Cgreen%28theta%2F2%29%29%2C%0D%0Alocate%289%2C0%2CA%29%2Clocate%28-0.1%2C7.25%2CB%29%2C%0D%0Alocate%28-0.1%2C0%2CC%29%2Clocate%28-0.3%2C3%2CO%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The cross sections of sphere and cone are<BR>\n" );
document.write( "a circle inscribed in an isosceles triangle.<BR>\n" );
document.write( "The center of the circle is the incenter of the triangle,<BR>\n" );
document.write( "at equal distance to all sides of the triangle,<BR>\n" );
document.write( "and on the bisectors of the triangle's angles.<BR>\n" );
document.write( "The height of the frustum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A3=6 ALIGN=MIDDLE  ALT=\"2%2A3=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the length of one side<BR>\n" );
document.write( "of a 3-4-5 right triangle with side length 6, 8, and 10.<BR>\n" );
document.write( "Its smaller angle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=BAC ALIGN=MIDDLE  ALT=\"theta=BAC\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29=6%2F10=0.6 ALIGN=MIDDLE  ALT=\"sin%28theta%29=6%2F10=0.6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29=8%2F10=0.8 ALIGN=MIDDLE  ALT=\"cos%28theta%29=8%2F10=0.8\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We are interested in right triangle AOC,<BR>\n" );
document.write( "because we want to find the length of AC.<BR>\n" );
document.write( "We know <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OC=3 ALIGN=MIDDLE  ALT=\"OC=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we know <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%2F2%29=OC%2FAC ALIGN=MIDDLE  ALT=\"tan%28theta%2F2%29=OC%2FAC\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The trigonometric identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%2F2%29=sin%28theta%29%2F%281%2Bcos%28theta%29%29 ALIGN=MIDDLE  ALT=\"tan%28theta%2F2%29=sin%28theta%29%2F%281%2Bcos%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "tells us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%2F2%29=0.6%2F%281%2B0.8%29=0.6%2F1.8=1%2F3 ALIGN=MIDDLE  ALT=\"tan%28theta%2F2%29=0.6%2F%281%2B0.8%29=0.6%2F1.8=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3=OC%2FAC=3%2FAC ALIGN=MIDDLE  ALT=\"1%2F3=OC%2FAC=3%2FAC\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC=9 ALIGN=MIDDLE  ALT=\"AC=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=9 ALIGN=MIDDLE  ALT=\"r=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the radius of the larger base of the frustum.<BR>\n" );
document.write( "Since right triangle ABC is similar to the 3-4-5 right triangle,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC%2FAC=BC%2F9=3%2F4 ALIGN=MIDDLE  ALT=\"BC%2FAC=BC%2F9=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=9%2A3%2F4=27%2F4=6.75 ALIGN=MIDDLE  ALT=\"BC=9%2A3%2F4=27%2F4=6.75\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is the height <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=27%2F4=6.75 ALIGN=MIDDLE  ALT=\"h=27%2F4=6.75\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the cone that the frustum came from.<BR>\n" );
document.write( "The volume of that cone is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=pi%281%2F3%29hr%5E2=pi%281%2F3%29%2A%2827%2F4%29%2A9%5E2=pi%289%2F4%29%2A9%5E2=%289%5E3%2F4%29pi ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The similar little cone lopped off to make the frustum has<BR>\n" );
document.write( "a base radius of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9-8=1 ALIGN=MIDDLE  ALT=\"9-8=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so it is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F9 ALIGN=MIDDLE  ALT=\"1%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > scale version of the original cone used to make the frustum.<BR>\n" );
document.write( "Consequently, the volume of that little cone is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%2A%281%2F9%29%5E3 ALIGN=MIDDLE  ALT=\"V%2A%281%2F9%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the volume of the frustum is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V-V%2A%281%2F9%29%5E3=V%2A%281-1%2F9%29%5E3=V%28%289%5E3-1%29%2F9%5E3%29=%289%5E3%2F4%29pi%2A%28%289%5E3-1%29%2F9%5E3%29%0D%0A=%289%5E3-1%29pi%2F4=%28729-1%29%2F4=728%2F4pi%2F4=182pi ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );