document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=831122>Question 831122</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A cylindrical jar of radius 3cm contains water to depth of 5cm. The water is then poured at a steady rate into an inverted conical container with its axis vertical. After t seconds, the depth of water in this container is x cm and the volume, V ml, of water that has been transferred is given by V= 1/3 (pie)(x)(x)(x).<BR>\n" );
document.write( "Given that all the water is transferred in 3 seconds,find<BR>\n" );
document.write( "a) dv/dt in terms of pie<BR>\n" );
document.write( "b) the rate at which c is increasing at the moment when x= 2.5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #501353 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=501353\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The volume in that cylindrical jar, calculated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2Aradius%5E2%2Aheight ALIGN=MIDDLE  ALT=\"pi%2Aradius%5E2%2Aheight\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A%283cm%29%5E2%2A%285cm%29=pi%2A45cm%5E3=pi%2A45mL ALIGN=MIDDLE  ALT=\"pi%2A%283cm%29%5E2%2A%285cm%29=pi%2A45cm%5E3=pi%2A45mL\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "a) If that volume is transferred at a steady rate in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > minutes, the average and constant rate is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdt=pi%2A45mL%2F%223+min%22=15pi ALIGN=MIDDLE  ALT=\"dV%2Fdt=pi%2A45mL%2F%223+min%22=15pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mL%2Fmin ALIGN=MIDDLE  ALT=\"mL%2Fmin\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15pi ALIGN=MIDDLE  ALT=\"15pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E3%2Fmin ALIGN=MIDDLE  ALT=\"cm%5E3%2Fmin\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).<BR>\n" );
document.write( " <BR>\n" );
document.write( "b) Since there is no \"c\" mentioned in the problem before part b),<BR>\n" );
document.write( "I assume that the question for part b) was<BR>\n" );
document.write( "to find the rate (maybe called c) at which the water level is increasing in the inverted conical container,<BR>\n" );
document.write( "when that water level there is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2.5cm ALIGN=MIDDLE  ALT=\"x=2.5cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That rate would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt ALIGN=MIDDLE  ALT=\"dx%2Fdt\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "At a constant rate of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15pi ALIGN=MIDDLE  ALT=\"15pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mL%2Fmin ALIGN=MIDDLE  ALT=\"mL%2Fmin\"   DISABLED_style= \"max-width:90%; height:auto;\" > , at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > minutes the volume in the inverted conical container would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=15pi%2At ALIGN=MIDDLE  ALT=\"V=15pi%2At\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We also know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=%281%2F3%29%2Api%2Ax%5E3 ALIGN=MIDDLE  ALT=\"V=%281%2F3%29%2Api%2Ax%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We can use those two equations to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt ALIGN=MIDDLE  ALT=\"dx%2Fdt\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt ALIGN=MIDDLE  ALT=\"dx%2Fdt\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2.5cm ALIGN=MIDDLE  ALT=\"x=2.5cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "I believe the easiest way to find the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt ALIGN=MIDDLE  ALT=\"dx%2Fdt\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2.5cm ALIGN=MIDDLE  ALT=\"x=2.5cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > is to consider that if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdt=%28dV%2Fdx%29%2A%28dx%2Fdt%29 ALIGN=MIDDLE  ALT=\"dV%2Fdt=%28dV%2Fdx%29%2A%28dx%2Fdt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdt=15pi ALIGN=MIDDLE  ALT=\"dV%2Fdt=15pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for all values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > ),<BR>\n" );
document.write( "and from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=%281%2F3%29%2Api%2Ax%5E3 ALIGN=MIDDLE  ALT=\"V=%281%2F3%29%2Api%2Ax%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > we calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx=pi%2Ax%5E2 ALIGN=MIDDLE  ALT=\"dV%2Fdx=pi%2Ax%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2.5cm ALIGN=MIDDLE  ALT=\"x=2.5cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx=pi%2A%282.5cm%29%5E2 ALIGN=MIDDLE  ALT=\"dV%2Fdx=pi%2A%282.5cm%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so at that point<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15pi ALIGN=MIDDLE  ALT=\"15pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E3%2Fmin=pi%2A%282.5cm%29%5E2%2A%28dx%2Fdt%29 ALIGN=MIDDLE  ALT=\"cm%5E3%2Fmin=pi%2A%282.5cm%29%5E2%2A%28dx%2Fdt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15cm%5E3%2Fmin=2.5%5E2cm%5E2%2A%28dx%2Fdt%29 ALIGN=MIDDLE  ALT=\"15cm%5E3%2Fmin=2.5%5E2cm%5E2%2A%28dx%2Fdt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt=15%2F2.5%5E2 ALIGN=MIDDLE  ALT=\"dx%2Fdt=15%2F2.5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%2Fmin ALIGN=MIDDLE  ALT=\"cm%2Fmin\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt=2.45 ALIGN=MIDDLE  ALT=\"dx%2Fdt=2.45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%2Fmin ALIGN=MIDDLE  ALT=\"cm%2Fmin\"   DISABLED_style= \"max-width:90%; height:auto;\" > at the point when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2.5cm ALIGN=MIDDLE  ALT=\"x=2.5cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "That does not give me a general expression for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt ALIGN=MIDDLE  ALT=\"dx%2Fdt\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but I do not believe the problem asked for that.<BR>\n" );
document.write( "However, just for fun, let's try to solve that way.<BR>\n" );
document.write( "With your excuse, I will skip writting the units this time.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28V=15pi%2At%2CV=%281%2F3%29%2Api%2Ax%5E3%29 ALIGN=MIDDLE  ALT=\"system%28V=15pi%2At%2CV=%281%2F3%29%2Api%2Ax%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15pi%2At=%281%2F3%29%2Api%2Ax%5E3 ALIGN=MIDDLE  ALT=\"15pi%2At=%281%2F3%29%2Api%2Ax%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15t=%281%2F3%29x%5E3 ALIGN=MIDDLE  ALT=\"15t=%281%2F3%29x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=45t=x%5E3 ALIGN=MIDDLE  ALT=\"45t=x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2845t%29%5E%221+%2F+3%22 ALIGN=MIDDLE  ALT=\"x=%2845t%29%5E%221+%2F+3%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt=45%5E%221+%2F+3%22%2A%281%2F3%29%2At%5E%22-+2+%2F+3%22 ALIGN=MIDDLE  ALT=\"dx%2Fdt=45%5E%221+%2F+3%22%2A%281%2F3%29%2At%5E%22-+2+%2F+3%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt=45%5E%221+%2F+3%22%2A%281%2F3%29%2Ft%5E%222+%2F+3%22 ALIGN=MIDDLE  ALT=\"dx%2Fdt=45%5E%221+%2F+3%22%2A%281%2F3%29%2Ft%5E%222+%2F+3%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That still does not look pretty, but I cannot make improve it too much, so I let it be.<BR>\n" );
document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2.5 ALIGN=MIDDLE  ALT=\"x=2.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=45t=2.5%5E3 ALIGN=MIDDLE  ALT=\"45t=2.5%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=2.5%5E3%2F45 ALIGN=MIDDLE  ALT=\"t=2.5%5E3%2F45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt=45%5E%221+%2F+3%22%2A%281%2F3%29%2At%5E%22-+2+%2F+3%22 ALIGN=MIDDLE  ALT=\"dx%2Fdt=45%5E%221+%2F+3%22%2A%281%2F3%29%2At%5E%22-+2+%2F+3%22\"   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=dx%2Fdt=45%5E%221+%2F+3%22%2A%281%2F3%29%2A%282.5%5E3%2F45%29%5E%22-+2+%2F+3%22 ALIGN=MIDDLE  ALT=\"dx%2Fdt=45%5E%221+%2F+3%22%2A%281%2F3%29%2A%282.5%5E3%2F45%29%5E%22-+2+%2F+3%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt=45%5E%221+%2F+3%22%2A%281%2F3%29%2A%2845%2F2.5%5E3%29%5E%222+%2F+3%22 ALIGN=MIDDLE  ALT=\"dx%2Fdt=45%5E%221+%2F+3%22%2A%281%2F3%29%2A%2845%2F2.5%5E3%29%5E%222+%2F+3%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt=45%5E%221+%2F+3%22%2A%281%2F3%29%2A45%5E%222+%2F+3%22%2F2.5%5E2 ALIGN=MIDDLE  ALT=\"dx%2Fdt=45%5E%221+%2F+3%22%2A%281%2F3%29%2A45%5E%222+%2F+3%22%2F2.5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt=45%2A%281%2F3%29%2F2.5%5E2 ALIGN=MIDDLE  ALT=\"dx%2Fdt=45%2A%281%2F3%29%2F2.5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt=15%2F2.5%5E2 ALIGN=MIDDLE  ALT=\"dx%2Fdt=15%2F2.5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt=2.45 ALIGN=MIDDLE  ALT=\"dx%2Fdt=2.45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So we get to the same result, and we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt ALIGN=MIDDLE  ALT=\"dx%2Fdt\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a bonus, in exchange for a bit more arithmetic work. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );