document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=636237>Question 636237</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use an inverted cone cut from the sphere (radius of 9cm) to dispense 100ml of a chemical. Chemical needs to be dispensed correct amount of quantity(100ml) and so 100ml increments need to be marked on the outside of cone. \r<BR>\n" );
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document.write( "The maximum volume of a cone is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32pi%2Aa%5E3%2F81 ALIGN=MIDDLE  ALT=\"32pi%2Aa%5E3%2F81\"   DISABLED_style= \"max-width:90%; height:auto;\" > and so the volume of this particular cone will be 904.7787cm^2. \r<BR>\n" );
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document.write( "I think ratios need to be used to find the distance between the increments but not sure how to do this if its right. Would be much appreciated if you would be able to help. THX. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #401059 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=401059\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=32pi%2Aa%5E3%2F81 ALIGN=MIDDLE  ALT=\"V=32pi%2Aa%5E3%2F81\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives you the maximum volume for a cone cut of a sphere of radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=9cm ALIGN=MIDDLE  ALT=\"a=9cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=904.7787cm%5E3 ALIGN=MIDDLE  ALT=\"V=904.7787cm%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The chemical filling up to the markings for 100-mL, 200-mL, 300-mL, and so on,<BR>\n" );
document.write( "would be in the shape of similar cones, all scaled down versions of the big cone.<BR>\n" );
document.write( "Their volumes (in cubic cm, which are equal to mL) would be directly proportional to the cube of the height in cm.<BR>\n" );
document.write( "The function relating height to volume for the chemical would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%28h%29=Kh%5E3 ALIGN=MIDDLE  ALT=\"V%28h%29=Kh%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so if the height of the cone (in cm) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H ALIGN=MIDDLE  ALT=\"H\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "we can calculate the heights (in cm) for the markings based on the proportion<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=K=V%2Fh%5E3=904.7787%2FH%5E3 ALIGN=MIDDLE  ALT=\"K=V%2Fh%5E3=904.7787%2FH%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E3=VH%5E3%2F904.7787 ALIGN=MIDDLE  ALT=\"h%5E3=VH%5E3%2F904.7787\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=root%283%2CVH%5E3%2F904.7787%29 ALIGN=MIDDLE  ALT=\"h=root%283%2CVH%5E3%2F904.7787%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That formula would work with heights or slant heights,<BR>\n" );
document.write( "because, in similar cones, they would be proportional.<BR>\n" );
document.write( "To get actual distances between markings on the outside of the cone,<BR>\n" );
document.write( "we need to get the slant height of the cone.<BR>\n" );
document.write( "There may be an easier way, but I will show how I calculated heights.<BR>\n" );
document.write( "Here is a cross section of the sphere (with center O), and the cone (with its axis in green):<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%2C%0D%0Acircle%280%2C0%2C9%29%2C%0D%0Atriangle%280%2C-9%2C8.4853%2C3%2C-8.4853%2C3%29%2C%0D%0Acircle%280%2C0%2C0.1%29%2Ccircle%280%2C0%2C0.2%29%2C%0D%0Agreen%28line%280%2C-10%2C0%2C10%29%29%2Clocate%280.1%2C4%2CD%29%2C%0D%0Alocate%280.1%2C-9%2CA%29%2Clocate%280.1%2C0%2CO%29%2C%0D%0Alocate%288.5%2C4%2CB%29%2Clocate%28-9%2C4%2CC%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The radius is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OA=OB=a ALIGN=MIDDLE  ALT=\"OA=OB=a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OD=ax ALIGN=MIDDLE  ALT=\"OD=ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > ia a fraction <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the radius.<BR>\n" );
document.write( "In the right triangle ODB, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DB%5E2%2BOD%5E2=OB%5E2 ALIGN=MIDDLE  ALT=\"DB%5E2%2BOD%5E2=OB%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DB%5E2%2B%28ax%29%5E2=a%5E2 ALIGN=MIDDLE  ALT=\"DB%5E2%2B%28ax%29%5E2=a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "allows to calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DB%5E2=a%5E2-%28ax%29%5E2=a%5E2-a%5E2x%5E2=a%5E2%281-x%5E2%29 ALIGN=MIDDLE  ALT=\"DB%5E2=a%5E2-%28ax%29%5E2=a%5E2-a%5E2x%5E2=a%5E2%281-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is the radius of the base of the cone, squared.<BR>\n" );
document.write( "The area of the base of the cone is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=pi%2Aa%5E2%281-x%5E2%29 ALIGN=MIDDLE  ALT=\"A=pi%2Aa%5E2%281-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The height of the cone is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H=OA%2BOD=a%2Bax=a%281%2Bx%29 ALIGN=MIDDLE  ALT=\"H=OA%2BOD=a%2Bax=a%281%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The volume of the cone <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=AH%2F3 ALIGN=MIDDLE  ALT=\"V=AH%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=pi%2Aa%5E2%281-x%5E2%29a%281%2Bx%29%2F3=%28pi%2Aa%5E3%2F3%29%281-x%5E2%29%281%2Bx%29=%28pi%2Aa%5E3%2F3%29%28-x%5E3-x%5E2%2Bx%2B1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using calculus, we can figure out that the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%28x%29 ALIGN=MIDDLE  ALT=\"V%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a maximum<BR>\n" );
document.write( "for <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;\" > ,<BR>\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-x%5E2%29%281%2Bx%29=%281-1%2F9%29%281%2B1%2F3%29=%288%2F9%29%284%2F3%29=32%2F27 ALIGN=MIDDLE  ALT=\"%281-x%5E2%29%281%2Bx%29=%281-1%2F9%29%281%2B1%2F3%29=%288%2F9%29%284%2F3%29=32%2F27\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5Bmax%5D=%28pi%2Aa%5E3%2F3%29%2832%2F27%29=32pi%2Aa%5E3%2F81 ALIGN=MIDDLE  ALT=\"V%5Bmax%5D=%28pi%2Aa%5E3%2F3%29%2832%2F27%29=32pi%2Aa%5E3%2F81\"   DISABLED_style= \"max-width:90%; height:auto;\" > as stated in your posted question.<BR>\n" );
document.write( "Most importantly, since <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;\" > ,<BR>\n" );
document.write( "the height of the cone (in cm) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H=AD=a%281%2Bx%29=9%281%2B1%2F3%29=9%284%2F3%29=12 ALIGN=MIDDLE  ALT=\"H=AD=a%281%2Bx%29=9%281%2B1%2F3%29=9%284%2F3%29=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The radius of the base of the cone, squared, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DB%5E2=a%5E2%281-x%5E2%29=9%5E2%281-1%2F9%29=9%5E2%288%2F9%29=72 ALIGN=MIDDLE  ALT=\"DB%5E2=a%5E2%281-x%5E2%29=9%5E2%281-1%2F9%29=9%5E2%288%2F9%29=72\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The slant height, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB ALIGN=MIDDLE  ALT=\"AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be calculated based on right triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABD ALIGN=MIDDLE  ALT=\"ABD\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB%5E2=AD%5E2%2BDB%5E2=12%5E2%2B72=144%2B72=216 ALIGN=MIDDLE  ALT=\"AB%5E2=AD%5E2%2BDB%5E2=12%5E2%2B72=144%2B72=216\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=sqrt%28216%29 ALIGN=MIDDLE  ALT=\"AB=sqrt%28216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14.6969cm ALIGN=MIDDLE  ALT=\"14.6969cm\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we can substitute the values for <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=AB ALIGN=MIDDLE  ALT=\"AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=root%283%2CV%2AH%5E3%2F904.7787%29 ALIGN=MIDDLE  ALT=\"h=root%283%2CV%2AH%5E3%2F904.7787%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or using the slant height, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB ALIGN=MIDDLE  ALT=\"AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > , of the cone,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=root%283%2CV%2AAB%5E3%2F904.7787%29 ALIGN=MIDDLE  ALT=\"d=root%283%2CV%2AAB%5E3%2F904.7787%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to calculate the height, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or the distance from the cone vertex, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the markings for volumes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" > of 100 mL, 200 mL, and so on.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=root%283%2CV%2A12%5E3%2F904.7787%29 ALIGN=MIDDLE  ALT=\"h=root%283%2CV%2A12%5E3%2F904.7787%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=root%283%2C1728V%2F904.7787%29 ALIGN=MIDDLE  ALT=\"h=root%283%2C1728V%2F904.7787%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=root%283%2CV%2A14.6969%5E3%2F904.7787%29 ALIGN=MIDDLE  ALT=\"d=root%283%2CV%2A14.6969%5E3%2F904.7787%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=root%283%2C3174.5387V%2F904.7787%29 ALIGN=MIDDLE  ALT=\"d=root%283%2C3174.5387V%2F904.7787%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );