document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183389>Question 1183389</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 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #813688 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=813688\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Solved a few days ago.  The response is copied below.<br><BR>\n" );
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document.write( "Let x be the inside length in meters of a side of the square base; let h be the height/depth of the tank.<br><BR>\n" );
document.write( "Then the volume of the tank is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2h=4000 ALIGN=MIDDLE  ALT=\"x%5E2h=4000\"   DISABLED_style= \"max-width:90%; height:auto;\" > [1]<br><BR>\n" );
document.write( "Since the thickness of the concrete is 50cm = 0.5m, the square base has dimensions (x+1)(x+1)(0.5).<br><BR>\n" );
document.write( "The four sides of the tank can be viewed as four congruent rectangular solids each with dimensions (x+0.5)(h)(0.5).<br><BR>\n" );
document.write( "So the total volume of the tank is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=0.5%28x%2B1%29%5E2%2B4%28x%2B0.5%29%28h%29%280.5%29 ALIGN=MIDDLE  ALT=\"V=0.5%28x%2B1%29%5E2%2B4%28x%2B0.5%29%28h%29%280.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > [2]<br><BR>\n" );
document.write( "Solve [1] for h in terms of x and substitute in the volume formula to get the volume in terms of the single variable x.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+4000%2Fx%5E2 ALIGN=MIDDLE  ALT=\"h+=+4000%2Fx%5E2\"   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=0.5%28x%2B1%29%5E2%2B4%28x%2B0.5%29%284000%2Fx%5E2%29%280.5%29 ALIGN=MIDDLE  ALT=\"V=0.5%28x%2B1%29%5E2%2B4%28x%2B0.5%29%284000%2Fx%5E2%29%280.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=0.5%28x%2B1%29%5E2%2B%282x%2B1%29%284000%2Fx%5E2%29 ALIGN=MIDDLE  ALT=\"V=0.5%28x%2B1%29%5E2%2B%282x%2B1%29%284000%2Fx%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=0.5%28x%2B1%29%5E2%2B8000%2Fx%2B4000%2Fx%5E2 ALIGN=MIDDLE  ALT=\"V=0.5%28x%2B1%29%5E2%2B8000%2Fx%2B4000%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Differentiate and set the derivative equal to zero to find the length of the side of the square base that minimizes the volume of concrete.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx=%28x%2B1%29-8000%2Fx%5E2-8000%2Fx%5E3 ALIGN=MIDDLE  ALT=\"dV%2Fdx=%28x%2B1%29-8000%2Fx%5E2-8000%2Fx%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%28x%2B1%29-8000x-8000+=+0 ALIGN=MIDDLE  ALT=\"x%5E3%28x%2B1%29-8000x-8000+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%28x%2B1%29-8000%28x%2B1%29+=+0 ALIGN=MIDDLE  ALT=\"x%5E3%28x%2B1%29-8000%28x%2B1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3-8000%29%28x%2B1%29+=+0 ALIGN=MIDDLE  ALT=\"%28x%5E3-8000%29%28x%2B1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3=8000 ALIGN=MIDDLE  ALT=\"x%5E3=8000\"   DISABLED_style= \"max-width:90%; height:auto;\" >  or  <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;\" ><br><BR>\n" );
document.write( "Clearly the negative solution makes no sense in the problem.  So<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3=8000 ALIGN=MIDDLE  ALT=\"x%5E3=8000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<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;\" ><br><BR>\n" );
document.write( "ANSWER: The volume of concrete to make the tank is minimized when the side length of the square base is 20m<br> </SPAN>\n" );
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