document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1101300>Question 1101300</A>: <I><SPAN STYLE=\"word-break: break-all;\"> iceman has enclosed one of his mortal enemies in a cube of ice. it is melting uniformly with the volume decreasing by 3 cubic feet per sec how fast is the surface area decreasing when the cubes edge is 5 ft?<BR>\n" );
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document.write( "i got lost when they are asking about the surface area. Thanks for helping out!<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #715962 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=715962\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> AS A CALCULUS PROBLEM:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" >= length of the edge, in ft<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=6s%5E2 ALIGN=MIDDLE  ALT=\"A=6s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >= surface area of the cube, in square feet<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=s%5E3 ALIGN=MIDDLE  ALT=\"V=s%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >= volume of the cube, in cubic feet<BR>\n" );
document.write( "<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, in seconds<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" > are functions 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( "We know the rate of change of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <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;\" > .<BR>\n" );
document.write( "If we figure out the relations among the derivatives of those functions,<BR>\n" );
document.write( "we can calculate the rate of change for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdt=6%2A2E%2A%28dE%2Fdt%29=12E%2A%28dE%2Fdt%29 ALIGN=MIDDLE  ALT=\"dA%2Fdt=6%2A2E%2A%28dE%2Fdt%29=12E%2A%28dE%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=dV%2Fdt=3E%5E2%2A%28dE%2Fdt%29=-3 ALIGN=MIDDLE  ALT=\"dV%2Fdt=3E%5E2%2A%28dE%2Fdt%29=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3E%5E2%2A%28dE%2Fdt%29=-3 ALIGN=MIDDLE  ALT=\"3E%5E2%2A%28dE%2Fdt%29=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dE%2Fdt=-1%2FE%5E2 ALIGN=MIDDLE  ALT=\"dE%2Fdt=-1%2FE%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting that into the expression for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdt ALIGN=MIDDLE  ALT=\"dA%2Fdt\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdt=12E%2A%28dE%2Fdt%29=12E%2A%28-1%2FE%5E2%29=-12%2FE ALIGN=MIDDLE  ALT=\"dA%2Fdt=12E%2A%28dE%2Fdt%29=12E%2A%28-1%2FE%5E2%29=-12%2FE\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E=5 ALIGN=MIDDLE  ALT=\"E=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdt=-12%2F5=-2.4 ALIGN=MIDDLE  ALT=\"dA%2Fdt=-12%2F5=-2.4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "When the cube's edge is 5 ft,<BR>\n" );
document.write( "the surface area is decreasing at a rate of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%282.4%29 ALIGN=MIDDLE  ALT=\"highlight%282.4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > square feet per second.<BR>\n" );
document.write( " <BR>\n" );
document.write( "OTHER OPTIONS:<BR>\n" );
document.write( "As a good approximation, calculate the average rate of area decrease<BR>\n" );
document.write( "between 0.01 seconds before the cube edge measures 5 ft<BR>\n" );
document.write( "and 0.01 seconds after the cube edge measures 5 ft.<BR>\n" );
document.write( "When the edge measures 5 ft, the volume, in cubic feet, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E3=125 ALIGN=MIDDLE  ALT=\"5%5E3=125\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "0.01 seconds before that, the volume, in cubic feet, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125%2B0.01%2A3=125.03 ALIGN=MIDDLE  ALT=\"125%2B0.01%2A3=125.03\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the edge length, in feet, is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C125.03%29=5.0004 ALIGN=MIDDLE  ALT=\"root%283%2C125.03%29=5.0004\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the surface area, in square feet, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2A5.0004%5E2=150.024 ALIGN=MIDDLE  ALT=\"6%2A5.0004%5E2=150.024\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "0.01 seconds after the edge measures 5 ft, the volume, in cubic feet, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125-0.01%2A3=124.97 ALIGN=MIDDLE  ALT=\"125-0.01%2A3=124.97\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the edge length, in feet, is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C124.97%29=4.9996 ALIGN=MIDDLE  ALT=\"root%283%2C124.97%29=4.9996\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the surface area, in square feet, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2A4.9996%5E2=149.976 ALIGN=MIDDLE  ALT=\"6%2A4.9996%5E2=149.976\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, over the course of 0.02 seconds,<BR>\n" );
document.write( "the surface area decrease, in square feet, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=150.024-149.976=0.048 ALIGN=MIDDLE  ALT=\"150.024-149.976=0.048\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "making the rate of decrease, in square feet per second,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.048%2F0.02=2.4 ALIGN=MIDDLE  ALT=\"0.048%2F0.02=2.4\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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