document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=356351>Question 356351</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how do i calculate the internal volume of a tapered bucket? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #254379 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=254379\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's look at a cone (in cross section). <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-10%2C10%2C-10%2C10%2Ccircle%280%2C-8%2C0.3%29%2Ccircle%28-5%2C8%2C0.3%29%2Ccircle%285%2C8%2C0.3%29%2Clocate%282%2C9%2Cr1%29%2Clocate%281%2C1%2Cr2%29%2Clocate%280.2%2C-2.5%2Ch1%29%2Clocate%280.2%2C4.5%2Ch2%29%2C%0D%0Aline%280%2C-8%2C5%2C8%29%2Cline%285%2C8%2C-5%2C8%29%2Ccircle%282.5%2C0%2C0.3%29%2Ccircle%28-2.5%2C0%2C0.3%29%2Cline%28-2.5%2C0%2C2.5%2C0%29%2Cline%28-5%2C8%2C0%2C-8%29%2C+line%280%2C-8%2C0%2C8%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Actually there are two cones in the picture.<BR>\n" );
document.write( "The large cone with a radius of r2 and height of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h1%2Bh2 ALIGN=MIDDLE  ALT=\"h1%2Bh2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and a smaller cone with a radius of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r1 ALIGN=MIDDLE  ALT=\"r1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and height <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h1 ALIGN=MIDDLE  ALT=\"h1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The figure in between would be the tapered bucket if you rotated the cross section about the axis.<BR>\n" );
document.write( "So the volume of the tapered bucket would be the volume of the large cone minus the volume of the small cone.<BR>\n" );
document.write( "The volume of a cone is,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5Bbc%5D=%28pi%2F3%29R%5E2%2AH ALIGN=MIDDLE  ALT=\"V%5Bbc%5D=%28pi%2F3%29R%5E2%2AH\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Big Cone:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5Bsc%5D=%28pi%2F3%29r2%5E2%2A%28h1%2Bh2%29 ALIGN=MIDDLE  ALT=\"V%5Bsc%5D=%28pi%2F3%29r2%5E2%2A%28h1%2Bh2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Small Cone:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=%28pi%2F3%29r1%5E2%2A%28h1%29 ALIGN=MIDDLE  ALT=\"V=%28pi%2F3%29r1%5E2%2A%28h1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Tapered Bucket:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5Btp%5D=%28pi%2F3%29r2%5E2%2A%28h1%2Bh2%29-%28pi%2F3%29r1%5E2%2A%28h1%29 ALIGN=MIDDLE  ALT=\"V%5Btp%5D=%28pi%2F3%29r2%5E2%2A%28h1%2Bh2%29-%28pi%2F3%29r1%5E2%2A%28h1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "There is also a relationship between the radii and the heights.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r1%29%2F%28h1%29=%28r2%29%2F%28h1%2Bh2%29 ALIGN=MIDDLE  ALT=\"%28r1%29%2F%28h1%29=%28r2%29%2F%28h1%2Bh2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r1%2Ah1%2Br1%2Ah2=r2%2Ah1 ALIGN=MIDDLE  ALT=\"r1%2Ah1%2Br1%2Ah2=r2%2Ah1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h1%28r2-r1%29=r1%2Ah2 ALIGN=MIDDLE  ALT=\"h1%28r2-r1%29=r1%2Ah2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h1=+%28r1%2Ah2%29%2F%28r2-r1%29+ ALIGN=MIDDLE  ALT=\"h1=+%28r1%2Ah2%29%2F%28r2-r1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h1%2Bh2=%28r1%2F%28r2-r1%29%29%2Bh2 ALIGN=MIDDLE  ALT=\"h1%2Bh2=%28r1%2F%28r2-r1%29%29%2Bh2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h1%2Bh2=%28r1%2F%28r2-r1%29%29h2%2Bh2%2A%28%28r2-r1%29%2F%28r2-r1%29%29 ALIGN=MIDDLE  ALT=\"h1%2Bh2=%28r1%2F%28r2-r1%29%29h2%2Bh2%2A%28%28r2-r1%29%2F%28r2-r1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h1%2Bh2=+%28r2%2F%28r2-r1%29%29%2Ah2 ALIGN=MIDDLE  ALT=\"h1%2Bh2=+%28r2%2F%28r2-r1%29%29%2Ah2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5Btp%5D=%28pi%2F3%29%28r2%5E2%2A%28h1%2Bh2%29-r1%5E2%2A%28h1%29%29 ALIGN=MIDDLE  ALT=\"V%5Btp%5D=%28pi%2F3%29%28r2%5E2%2A%28h1%2Bh2%29-r1%5E2%2A%28h1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5Btp%5D=%28pi%2F3%29%28r2%5E2%2A%28%28r2%2F%28r2-r1%29%29%2Ah2%29-r1%5E2%2A%28%28r1%2Ah2%29%2F%28r2-r1%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5Btp%5D=%28pi%2F3%29%28h2%2F%28r2-r1%29%29%28r2%5E3-r1%5E3%29%29 ALIGN=MIDDLE  ALT=\"V%5Btp%5D=%28pi%2F3%29%28h2%2F%28r2-r1%29%29%28r2%5E3-r1%5E3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Dividing by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r2-r1%29 ALIGN=MIDDLE  ALT=\"%28r2-r1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28V%5Btp%5D=%28pi%2F3%29%28h2%29%28r2%5E2%2Br1%2Ar2%2Br1%5E2%29%29%29 ALIGN=MIDDLE  ALT=\"highlight%28V%5Btp%5D=%28pi%2F3%29%28h2%29%28r2%5E2%2Br1%2Ar2%2Br1%5E2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );