document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1106186>Question 1106186</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Water exits a conical tank at a constant rate of o.2 m3/minutes. If the surface of the water has radius r:<BR>\n" );
document.write( "a) find V(r), the volume of the water remaining<BR>\n" );
document.write( "b) find the rate at which the surface radius is changing at the instant when the water is 5m deep.<BR>\n" );
document.write( "The diameter is 6m. Height 8m.  </SPAN>\n" );
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document.write( "The conical tank has a radius of 3m and a height (depth) of 8m.  The ratio of the radius of the surface of the water to the depth of the water will always be 3:8.  So<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%283%2F8%29h ALIGN=MIDDLE  ALT=\"r+=+%283%2F8%29h\"   DISABLED_style= \"max-width:90%; height:auto;\" >; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+%288%2F3%29r ALIGN=MIDDLE  ALT=\"h+=+%288%2F3%29r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(a) The volume remaining at any time is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%281%2F3%29%28pi%29%28r%5E2%29%28h%29+=+%281%2F3%29%28pi%29%28r%5E2%29%28%288%2F3%29r%29+=+%288%2F9%29%28pi%29%28r%5E3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%28r%29+=+%288%2F9%29%28pi%29%28r%5E3%29 ALIGN=MIDDLE  ALT=\"V%28r%29+=+%288%2F9%29%28pi%29%28r%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(b) We want to find the rate at which the radius of the surface of the water is changing (dr/dt) when the depth is 5m.  We are given that the volume of water in the tank is decreasing at a rate of 0.2 m^3/minute: dV/dt = -0.2.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdt+=+%28dV%2Fdr%29%2A%28dr%2Fdt%29 ALIGN=MIDDLE  ALT=\"dV%2Fdt+=+%28dV%2Fdr%29%2A%28dr%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=dr%2Fdt+=+%28dV%2Fdt%29%2F%28dV%2Fdr%29 ALIGN=MIDDLE  ALT=\"dr%2Fdt+=+%28dV%2Fdt%29%2F%28dV%2Fdr%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We know dV/dt; and from part (a) we can find dV/dr to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288%2F3%29%28pi%29%28r%5E2%29 ALIGN=MIDDLE  ALT=\"%288%2F3%29%28pi%29%28r%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dr%2Fdt+=+%28-0.2%29%2F%28%288%2F3%29%28pi%29%28r%5E2%29%29 ALIGN=MIDDLE  ALT=\"dr%2Fdt+=+%28-0.2%29%2F%28%288%2F3%29%28pi%29%28r%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "When the depth is 5m, the radius is (3/8)*5 = (15/8)m.  So dr/dt when the depth is 5 is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dr%2Fdt+=+%28-0.2%29%2F%28%288%2F3%29%28pi%29%28%2815%2F8%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"dr%2Fdt+=+%28-0.2%29%2F%28%288%2F3%29%28pi%29%28%2815%2F8%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dr%2Fdt+=+%28-1%2F5%29%2F%28%2875%2F8%29%28pi%29%29 ALIGN=MIDDLE  ALT=\"dr%2Fdt+=+%28-1%2F5%29%2F%28%2875%2F8%29%28pi%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dr%2Fdt+=+%28-8%29%2F%28375%28pi%29%29 ALIGN=MIDDLE  ALT=\"dr%2Fdt+=+%28-8%29%2F%28375%28pi%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );