document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=745144>Question 745144</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dry sand is being poured into a conical pile at a rate of 10 cubic meters per minute; the diameter of the pile is equal to the height. At what rate is the height of the cone changing when there are 144&#960; cubic meters of sand in the pile? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #453763 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=453763\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" >= height of the cone = diameter of the base<BR>\n" );
document.write( "The volume of a cone, <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 calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=%281%2F3%29%2AB%2Ah=%281%2F3%29%2Api%2Ar%5E2%2Ah ALIGN=MIDDLE  ALT=\"V=%281%2F3%29%2AB%2Ah=%281%2F3%29%2Api%2Ar%5E2%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" > based on the height and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" >= area of the base, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" >= radius of the base<BR>\n" );
document.write( "In the case of the pile of sand, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=h%2F2 ALIGN=MIDDLE  ALT=\"r=h%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=%281%2F3%29%2Api%2A%28h%2F2%29%5E2%2Ah=%281%2F3%29%2Api%2A%28h%5E2%2F4%29%2Ah=%281%2F12%29%2Api%2Ah%5E3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=144pi ALIGN=MIDDLE  ALT=\"V=144pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F12%29%2Api%2Ah%5E3=144pi ALIGN=MIDDLE  ALT=\"%281%2F12%29%2Api%2Ah%5E3=144pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E3=144%2A12=12%5E3 ALIGN=MIDDLE  ALT=\"h%5E3=144%2A12=12%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=12 ALIGN=MIDDLE  ALT=\"h=12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "For all other times, solving 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;\" > we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=%281%2F12%29%2Api%2Ah%5E3 ALIGN=MIDDLE  ALT=\"V=%281%2F12%29%2Api%2Ah%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12V%2Fpi=h%5E3 ALIGN=MIDDLE  ALT=\"12V%2Fpi=h%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=root%283%2C12V%2Fpi%29 ALIGN=MIDDLE  ALT=\"h=root%283%2C12V%2Fpi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=%2812V%2Fpi%29%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"h=%2812V%2Fpi%29%5E%281%2F3%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 ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a linear function of time<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > time after <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=144pi ALIGN=MIDDLE  ALT=\"V=144pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > (with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" > in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E3 ALIGN=MIDDLE  ALT=\"m%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > in minutes,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=144pi%2B10t ALIGN=MIDDLE  ALT=\"V=144pi%2B10t\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=%2812%2A%28144pi%2B10t%29%2Fpi%29%5E%281%2F3%29=%28%281728pi%2B120t%29%2Fpi%29%5E%281%2F3%29=%281728%2B120t%2Fpi%29%5E%281%2F3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not a linear function of time, the rate of change 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;\" > changes with time, and for an exact value we have to calculate it using calculus.<BR>\n" );
document.write( "Without calculus, we can get approximate values.<BR>\n" );
document.write( " <BR>\n" );
document.write( "WITH CALCULUS:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dh%2Fdt=%281%2F3%29%281728%2B120t%2Fpi%29%5E%28-2%2F3%29%2A%28120%2Fpi%29 ALIGN=MIDDLE  ALT=\"dh%2Fdt=%281%2F3%29%281728%2B120t%2Fpi%29%5E%28-2%2F3%29%2A%28120%2Fpi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "and at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dh%2Fdt=%281%2F3%29%2A%281728%2B0%2Fpi%29%5E%28-2%2F3%29%2A%28120%2Fpi%29=%281%2F3%29%2A%281%2F12%5E2%29%2A%28120%2Fpi%29=40%2F144pi=highlight%285%2F18pi%29=highlight%28about0.088419%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "WITHOUT CALCULUS:<BR>\n" );
document.write( "We can get estimates of the instantaneous rate of change in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=144pi ALIGN=MIDDLE  ALT=\"V=144pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > by calculating the average rate of change over short periods of time, when <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 about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=144pi ALIGN=MIDDLE  ALT=\"144pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "For example, between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0.1 ALIGN=MIDDLE  ALT=\"t=0.1\"   DISABLED_style= \"max-width:90%; height:auto;\" > minutes, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" > changes from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%280%29=144pi ALIGN=MIDDLE  ALT=\"V%280%29=144pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%280.1%29=144pi%2B1 ALIGN=MIDDLE  ALT=\"V%280.1%29=144pi%2B1\"   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;\" > changes from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%280%29=12 ALIGN=MIDDLE  ALT=\"h%280%29=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%280.1%29=%281728%2B12%2Fpi%29%5E%281%2F3%29=about12.00883543 ALIGN=MIDDLE  ALT=\"h%280.1%29=%281728%2B12%2Fpi%29%5E%281%2F3%29=about12.00883543\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The average rate of change is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.00883543%2F0.1=0.0883543 ALIGN=MIDDLE  ALT=\"0.00883543%2F0.1=0.0883543\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If we use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0.01 ALIGN=MIDDLE  ALT=\"t=0.01\"   DISABLED_style= \"max-width:90%; height:auto;\" > with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%280%29=12 ALIGN=MIDDLE  ALT=\"h%280%29=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%280.01%29=%281728%2B1.2%2Fpi%29%5E%281%2F3%29=about12.00088413 ALIGN=MIDDLE  ALT=\"h%280.01%29=%281728%2B1.2%2Fpi%29%5E%281%2F3%29=about12.00088413\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "we get an average rate of change of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.00088413%2F0.01=0.88413 ALIGN=MIDDLE  ALT=\"0.00088413%2F0.01=0.88413\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Either way, we see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%280.884%29 ALIGN=MIDDLE  ALT=\"highlight%280.884%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > meters per minute is a good estimate of the rate of increase for the height of the pile when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=144pi ALIGN=MIDDLE  ALT=\"V=144pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "We did not really need to keep using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=%281%2F12%29%2Api%2Ah%5E3 ALIGN=MIDDLE  ALT=\"V=%281%2F12%29%2Api%2Ah%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=%2812V%2Fpi%29%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"h=%2812V%2Fpi%29%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=%281728%2B120t%2Fpi%29%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"h=%281728%2B120t%2Fpi%29%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to calculate an approximate rate of change without calculus. All we needed to know is that for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=144pi ALIGN=MIDDLE  ALT=\"V=144pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=12 ALIGN=MIDDLE  ALT=\"h=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "We know that the ratio of volumes of similar solids is the cube of the ratio of lengths for any dimension measured, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%28t%29%2FV%280%29=%28h%28t%29%2Fh%280%29%29%5E3 ALIGN=MIDDLE  ALT=\"V%28t%29%2FV%280%29=%28h%28t%29%2Fh%280%29%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29%2Fh%280%29=%28V%28t%29%2FV%280%29%29%5E%281%2F3%29=root%283%2CV%28t%29%2FV%280%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That would let us calculate the height for any other volume<BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%280.1%29=144pi%2B1 ALIGN=MIDDLE  ALT=\"V%280.1%29=144pi%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%280.1%29%2Fh%280%29=root%283%2C%28144pi%2B1%29%2F144%29=root%283%2C1.002210485%29=0.000736286 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%280.1%29=V%280%29%2A0.000736286=12%2A0.000736286=12.00883543 ALIGN=MIDDLE  ALT=\"V%280.1%29=V%280%29%2A0.000736286=12%2A0.000736286=12.00883543\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and the average rate of change<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28h%280.1%29-h%280%29%29%2F%280.1-0%29=%2812.00883543-12%29%2F0.1=0.0883543 ALIGN=MIDDLE  ALT=\"%28h%280.1%29-h%280%29%29%2F%280.1-0%29=%2812.00883543-12%29%2F0.1=0.0883543\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );