document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183388>Question 1183388</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An open lid tank to be made by concrete has width 50𝑐𝑚, inside capacity of 4000 𝑚3<BR>\n" );
document.write( "and square base. Find the inner dimension of the tank with the minimum volume of <BR>\n" );
document.write( "concrete.<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 #813692 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=813692\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The thickness of the open-lid tank is 50 cm = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29+m ALIGN=MIDDLE  ALT=\"%281%2F2%29+m\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Let the interior square base have a side of length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+ ALIGN=MIDDLE  ALT=\"+x+\"   DISABLED_style= \"max-width:90%; height:auto;\" > meters.  \r<BR>\n" );
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document.write( "Also let the inner height be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+ ALIGN=MIDDLE  ALT=\"y+\"   DISABLED_style= \"max-width:90%; height:auto;\" > m.  <BR>\n" );
document.write( "This then gives \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Ay+=+4000 ALIGN=MIDDLE  ALT=\"x%5E2%2Ay+=+4000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "since it is given that the inside capacity is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4000+m%5E3 ALIGN=MIDDLE  ALT=\"4000+m%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "For the exterior of the tank, the square base has a side of length <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+%2B+2%2A%281%2F2%29+=+x%2B1 ALIGN=MIDDLE  ALT=\"+x+%2B+2%2A%281%2F2%29+=+x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > meters, and a height of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%2B+1%2F2 ALIGN=MIDDLE  ALT=\"y+%2B+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > meters.\r<BR>\n" );
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document.write( "The external volume is then \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%28x%2B1%29%5E2%2A%28y%2B1%2F2%29+=+%28x%2B1%29%5E2%2A%28+4000%2Fx%5E2+%2B1%2F2%29 ALIGN=MIDDLE  ALT=\"V+=+%28x%2B1%29%5E2%2A%28y%2B1%2F2%29+=+%28x%2B1%29%5E2%2A%28+4000%2Fx%5E2+%2B1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, after substituting for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the previous equation.\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx+=+2%28x%2B1%29%284000%2Fx%5E2%2B1%2F2%29-%28x%2B1%29%5E2%288000%2Fx%5E3%29%0D%0A%0D%0A=+%28x%2B1%29%288000%2Fx%5E2%2B1%29-%28x%2B1%29%5E2%288000%2Fx%5E3%29+=+%28%28x%2B1%29%2A%28x%5E3+-+8000%29%29%2Fx%5E3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "after simplifying the expression on the right side.\r<BR>\n" );
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document.write( "Setting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx+=+%28%28x%2B1%29%2A%28x%5E3+-+8000%29%29%2Fx%5E3+=+0 ALIGN=MIDDLE  ALT=\"dV%2Fdx+=+%28%28x%2B1%29%2A%28x%5E3+-+8000%29%29%2Fx%5E3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+20+m ALIGN=MIDDLE  ALT=\"x+=+20+m\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "We cannot accept <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1 ALIGN=MIDDLE  ALT=\"x+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > since the domain of V is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E+0 ALIGN=MIDDLE  ALT=\"x+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now if  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C+20 ALIGN=MIDDLE  ALT=\"x+%3C+20\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx+%3C+0+ ALIGN=MIDDLE  ALT=\"dV%2Fdx+%3C+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and \r<BR>\n" );
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document.write( "when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E+20 ALIGN=MIDDLE  ALT=\"x+%3E+20\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx+%3E+0+ ALIGN=MIDDLE  ALT=\"dV%2Fdx+%3E+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  so by the the 1st derivative test, \r<BR>\n" );
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document.write( "there is a local minimum at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+20 ALIGN=MIDDLE  ALT=\"x+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since it is the only critical \r<BR>\n" );
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document.write( "point in the domain (0, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >), the minimum at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+20 ALIGN=MIDDLE  ALT=\"x+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also\r<BR>\n" );
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document.write( "absolute minimum. \r<BR>\n" );
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document.write( "Therefore the inner dimensions of the tank are 20 m x 20 m x 10m. (The height is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+4000%2Fx%5E2+=+4000%2F20%5E2+=+10+m ALIGN=MIDDLE  ALT=\"y+=+4000%2Fx%5E2+=+4000%2F20%5E2+=+10+m\"   DISABLED_style= \"max-width:90%; height:auto;\" >.)\r<BR>\n" );
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