document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197602>Question 1197602</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Water is pouring into an inverted cone at the rate of 8 cubic feet per minute. If the height of the cone is 12 ft and the radius of its base is 6 ft, how fast is the water level rising when the water is 4 ft deep? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830937 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=830937\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The volume is increasing at a rate of 8 cubic feet per minute; we want to know how fast the water level (height) is changing when the height is 8 feet.<br><BR>\n" );
document.write( "Since we are given the rate of change of the volume and we want to find the rate of change of the height, we need a formula for the volume in terms of the height.<br><BR>\n" );
document.write( "Volume of a cone: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28base%29%28height%29=%281%2F3%29%28pi%2Ar%5E2%29%28h%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28base%29%28height%29=%281%2F3%29%28pi%2Ar%5E2%29%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The height of the cone is 12 feet and the radius of its base is 6 feet.  Since the sides of the cone are straight, the ratio 6/12 = 1/2 of radius to height is constant.  We want our formula in terms of height, so<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=%281%2F2%29h ALIGN=MIDDLE  ALT=\"r=%281%2F2%29h\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And the volume of the cone in terms of the height is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=%281%2F3%29%28pi%2A%28h%2F2%29%5E2%29%28h%29=%28pi%2F12%29h%5E3 ALIGN=MIDDLE  ALT=\"V=%281%2F3%29%28pi%2A%28h%2F2%29%5E2%29%28h%29=%28pi%2F12%29h%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we are ready to use our calculus for this related rates problem.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdt=%28pi%2F12%29%283h%5E2%29%28dh%2Fdt%29 ALIGN=MIDDLE  ALT=\"dV%2Fdt=%28pi%2F12%29%283h%5E2%29%28dh%2Fdt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdt=%28pi%2F4%29%28h%5E2%29%28dh%2Fdt%29 ALIGN=MIDDLE  ALT=\"dV%2Fdt=%28pi%2F4%29%28h%5E2%29%28dh%2Fdt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dh%2Fdt=%284%2F%28%28h%5E2%29pi%29%29%28dV%2Fdt%29 ALIGN=MIDDLE  ALT=\"dh%2Fdt=%284%2F%28%28h%5E2%29pi%29%29%28dV%2Fdt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We want the rate of change of the height, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dh%2Fdt ALIGN=MIDDLE  ALT=\"dh%2Fdt\"   DISABLED_style= \"max-width:90%; height:auto;\" >, when 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;\" > is 4, knowing that the rate of change in the volume, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdt ALIGN=MIDDLE  ALT=\"dV%2Fdt\"   DISABLED_style= \"max-width:90%; height:auto;\" >, is 8:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dh%2Fdt=%284%2F%28%28%284%29%5E2%29pi%29%29%288%29=32%2F%2816pi%29=2%2Fpi ALIGN=MIDDLE  ALT=\"dh%2Fdt=%284%2F%28%28%284%29%5E2%29pi%29%29%288%29=32%2F%2816pi%29=2%2Fpi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: The water level is rising at a rate of (2/pi) feet per second when the water is 4 feet deep.<br>\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );