document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192719>Question 1192719</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Sand is being poured at a rate of 0.50 m^3/min onto a conical pile whose radius is always equal to 2/3 of its height. At one point in time, an observer notices that the height of the sand pile is increasing at a rate of 8.0 cm/min. What was the height of the pile at this instant? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824625 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=824625\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "r = radius in meters<BR>\n" );
document.write( "h = height in meters<BR>\n" );
document.write( "t = time in minutes\r<BR>\n" );
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document.write( "dr/dt = change in radius over change in time, ie speed at which the radius changes<BR>\n" );
document.write( "dh/dt = change in height over change in time, ie speed at which the height changes<BR>\n" );
document.write( "Each speed is measured in meters per minute.\r<BR>\n" );
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document.write( "100 cm = 1 m<BR>\n" );
document.write( "8.0 cm/min = (8.0 cm/min)*(1 m/100 cm) = 0.08 meters per minute\r<BR>\n" );
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document.write( "The sentence<BR>\n" );
document.write( "\"height of the sand pile is increasing at a rate of 8.0 cm/min\" <BR>\n" );
document.write( "is equivalent to<BR>\n" );
document.write( "\"height of the sand pile is increasing at a rate of 0.08 m/min\". <BR>\n" );
document.write( "This is the dh/dt value for this snapshot in time.\r<BR>\n" );
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document.write( "r = (2/3)h since the radius is 2/3 the height<BR>\n" );
document.write( "That rearranges into<BR>\n" );
document.write( "3r = 2h<BR>\n" );
document.write( "h = 3r/2<BR>\n" );
document.write( "h = 1.5r\r<BR>\n" );
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document.write( "Apply the derivative with respect to t to both sides<BR>\n" );
document.write( "h = (3/2)r<BR>\n" );
document.write( "dh/dt = (3/2)*dr/dt<BR>\n" );
document.write( "dh/dt = (3/2)*dr/dt\r<BR>\n" );
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document.write( "Plug in dh/dt = 0.08 mentioned earlier and solve for dr/dt<BR>\n" );
document.write( "dh/dt = (3/2)*dr/dt<BR>\n" );
document.write( "dr/dt = (2/3)*dh/dt<BR>\n" );
document.write( "dr/dt = (2/3)*0.08<BR>\n" );
document.write( "dr/dt = (2/3)*(8/100)<BR>\n" );
document.write( "dr/dt = 16/300<BR>\n" );
document.write( "dr/dt = 4/75\r<BR>\n" );
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document.write( "We have this so far:<BR>\n" );
document.write( "h = 3r/2 = 1.5r<BR>\n" );
document.write( "dr/dt = 4/75\r<BR>\n" );
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document.write( "Recall the volume of a cone with radius r and height h is <BR>\n" );
document.write( "V = (1/3)*pi*r^2*h\r<BR>\n" );
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document.write( "Subsitution allows us to say<BR>\n" );
document.write( "V = (1/3)*pi*r^2*h<BR>\n" );
document.write( "V = (1/3)*pi*r^2*(3r/2)<BR>\n" );
document.write( "V = (1/3)*(3/2)*pi*r^3<BR>\n" );
document.write( "V = (1/2)*pi*r^3\r<BR>\n" );
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document.write( "Apply the derivative with respect to t<BR>\n" );
document.write( "V = (1/2)*pi*r^3<BR>\n" );
document.write( "dV/dt = d/dt[ (1/2)*pi*r^3 ]<BR>\n" );
document.write( "dV/dt = (1/2)*pi*d/dt[ r^3 ]<BR>\n" );
document.write( "dV/dt = (1/2)*pi*(3r^2 * dr/dt)<BR>\n" );
document.write( "Don't forget about the chain rule.<BR>\n" );
document.write( "The radius r is a function of t. \r<BR>\n" );
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document.write( "We'll plug in the rate of change of the volume dV/dt = 0.50 m^3/min along with the rate of change in the radius dr/dt = 4/75<BR>\n" );
document.write( "Then we isolate r.<BR>\n" );
document.write( "dV/dt = (1/2)*pi*(3r^2 * dr/dt)<BR>\n" );
document.write( "0.50 = (1/2)*pi*(3r^2 * 4/75)<BR>\n" );
document.write( "0.50 = (2/25)*pi*r^2<BR>\n" );
document.write( "(2/25)*pi*r^2 = 0.50<BR>\n" );
document.write( "pi*r^2 = 0.50*(25/2)<BR>\n" );
document.write( "pi*r^2 = 6.25<BR>\n" );
document.write( "r^2 = (6.25)/pi<BR>\n" );
document.write( "r = sqrt(6.25/pi)\r<BR>\n" );
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document.write( "This leads to<BR>\n" );
document.write( "h = (3/2)r<BR>\n" );
document.write( "h = (3/2)sqrt(6.25/pi)<BR>\n" );
document.write( "which is the exact height in meters.\r<BR>\n" );
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document.write( "This approximates to (3/2)sqrt(6.25/pi) = 2.1157109 meters<BR>\n" );
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