document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=164817>Question 164817</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have to find the least expensive cost for a steel frame building which measures 600 cu. meters.  The prices of horizontal beams are 20d and 30w, vertical beams are 50h.  Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #121505 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=121505\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The cost equation is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=20D%2B30W%2B50H ALIGN=MIDDLE  ALT=\"C=20D%2B30W%2B50H\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You also know that <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=D%2AW%2AH=600 ALIGN=MIDDLE  ALT=\"V=D%2AW%2AH=600\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can replace one of the variables in the cost equation.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=600%2F%28WH%29 ALIGN=MIDDLE  ALT=\"D=600%2F%28WH%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then the cost equation becomes,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=20%28600%2F%28WH%29%29%2B30W%2B50H ALIGN=MIDDLE  ALT=\"C=20%28600%2F%28WH%29%29%2B30W%2B50H\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=%2812000%29%2F%28WH%29%2B30W%2B50H ALIGN=MIDDLE  ALT=\"C=%2812000%29%2F%28WH%29%2B30W%2B50H\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=12000W%5E%28-1%29H%5E%28-1%29%2B30W%2B50H ALIGN=MIDDLE  ALT=\"C=12000W%5E%28-1%29H%5E%28-1%29%2B30W%2B50H\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To minimize the cost function we have to find the partial derivative of C with respect to W and H and set those equal to zero.<BR>\n" );
document.write( "1.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dC%2FdW=12000H%5E%28-1%29%28-W%5E%28-2%29%29%2B30=0 ALIGN=MIDDLE  ALT=\"dC%2FdW=12000H%5E%28-1%29%28-W%5E%28-2%29%29%2B30=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dC%2FdH=12000W%5E%28-1%29%28-H%5E%28-2%29%29%2B50=0 ALIGN=MIDDLE  ALT=\"dC%2FdH=12000W%5E%28-1%29%28-H%5E%28-2%29%29%2B50=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The understanding is that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dC%2FdW ALIGN=MIDDLE  ALT=\"dC%2FdW\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dC%2FdH ALIGN=MIDDLE  ALT=\"dC%2FdH\"   DISABLED_style= \"max-width:90%; height:auto;\" > are partial derivatives.<BR>\n" );
document.write( "From eq. 1,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12000H%5E%28-1%29%28-W%5E%28-2%29%29%2B30=0 ALIGN=MIDDLE  ALT=\"12000H%5E%28-1%29%28-W%5E%28-2%29%29%2B30=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12000H%5E%28-1%29%28W%5E%28-2%29%29=30 ALIGN=MIDDLE  ALT=\"12000H%5E%28-1%29%28W%5E%28-2%29%29=30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "3.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=HW%5E2=400 ALIGN=MIDDLE  ALT=\"HW%5E2=400\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From eq. 2,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12000W%5E%28-1%29%28-H%5E%28-2%29%29%2B50=0 ALIGN=MIDDLE  ALT=\"12000W%5E%28-1%29%28-H%5E%28-2%29%29%2B50=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12000W%5E%28-1%29%28H%5E%28-2%29%29=50 ALIGN=MIDDLE  ALT=\"12000W%5E%28-1%29%28H%5E%28-2%29%29=50\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "4.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%5E2W=240 ALIGN=MIDDLE  ALT=\"H%5E2W=240\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From eq. 4, <BR>\n" );
document.write( "4.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%5E2W=240 ALIGN=MIDDLE  ALT=\"H%5E2W=240\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W=240%2FH%5E2 ALIGN=MIDDLE  ALT=\"W=240%2FH%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W%5E2=57600%2FH%5E4 ALIGN=MIDDLE  ALT=\"W%5E2=57600%2FH%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substitute this expression in eq. 3 and solve for H,<BR>\n" );
document.write( "3.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=HW%5E2=400 ALIGN=MIDDLE  ALT=\"HW%5E2=400\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%2857600%2FH%5E4%29=400 ALIGN=MIDDLE  ALT=\"H%2857600%2FH%5E4%29=400\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=57600%2FH%5E3=400 ALIGN=MIDDLE  ALT=\"57600%2FH%5E3=400\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%5E3=57600%2F400=144 ALIGN=MIDDLE  ALT=\"H%5E3=57600%2F400=144\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H=5.24 ALIGN=MIDDLE  ALT=\"H=5.24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From eq. 4,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W=240%2FH%5E2 ALIGN=MIDDLE  ALT=\"W=240%2FH%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W=240%2F%285.24%29%5E2 ALIGN=MIDDLE  ALT=\"W=240%2F%285.24%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W=240%2F%2827.4576%29 ALIGN=MIDDLE  ALT=\"W=240%2F%2827.4576%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W=8.74 ALIGN=MIDDLE  ALT=\"W=8.74\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Finally,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=600%2F%28WH%29 ALIGN=MIDDLE  ALT=\"D=600%2F%28WH%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=600%2F%285.24%2A8.74%29 ALIGN=MIDDLE  ALT=\"D=600%2F%285.24%2A8.74%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=600%2F45.7976 ALIGN=MIDDLE  ALT=\"D=600%2F45.7976\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=13.10 ALIGN=MIDDLE  ALT=\"D=13.10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The total cost would then be,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=20%2813.10%29%2B30%2A%288.74%29%2B50%2A%285.24%29 ALIGN=MIDDLE  ALT=\"C=20%2813.10%29%2B30%2A%288.74%29%2B50%2A%285.24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=262%2B262.2%2B262 ALIGN=MIDDLE  ALT=\"C=262%2B262.2%2B262\"   DISABLED_style= \"max-width:90%; height:auto;\" >.... An interesting finding, minimizing cost means each portion makes up exactly 1/3 of the cost. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=786.2 ALIGN=MIDDLE  ALT=\"C=786.2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );