document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1094699>Question 1094699</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a rectangular block of iron has to be cast so that its length is three times its width and its volume is a maximun,.find the dimensions of the block if its total surface area is not to exceed 1800 square cm </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #709374 by </B><A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><B>ankor@dixie-net.com(22740)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><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=ankor@dixie-net.com><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=709374\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  rectangular block of iron has to be cast so that its length is three times its width and its volume is a maximun,.<BR>\n" );
document.write( "find the dimensions of the block if its total surface area is not to exceed 1800 square cm <BR>\n" );
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document.write( "let L = the length of block<BR>\n" );
document.write( "let x = the width<BR>\n" );
document.write( "let h = the height<BR>\n" );
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document.write( "The surface area equation<BR>\n" );
document.write( "2Lx + 2Lh + 2xh = 1800<BR>\n" );
document.write( "simplify divide by 2<BR>\n" );
document.write( " xL + Lh + xh = 900 <BR>\n" );
document.write( "we know that L = 3x<BR>\n" );
document.write( "3x^2 + 3xh + xh = 900<BR>\n" );
document.write( "3x^2 + 4xh = 900<BR>\n" );
document.write( "4xh = 900 - 3x^2<BR>\n" );
document.write( "h = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28900-3x%5E2%29%29%2F%284x%29 ALIGN=MIDDLE  ALT=\"%28%28900-3x%5E2%29%29%2F%284x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "The volume equation<BR>\n" );
document.write( "V = L * x * h<BR>\n" );
document.write( "Replace L with 3x<BR>\n" );
document.write( "V = 3x * x * h<BR>\n" );
document.write( "V = 3x^2*h<BR>\n" );
document.write( "Replace h with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28900-3x%5E2%29%29%2F%284x%29 ALIGN=MIDDLE  ALT=\"%28%28900-3x%5E2%29%29%2F%284x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "V = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%5E2%28900-3x%5E2%29%29%2F%284x%29 ALIGN=MIDDLE  ALT=\"%283x%5E2%28900-3x%5E2%29%29%2F%284x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "cancel x<BR>\n" );
document.write( "V = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%28900-3x%5E2%29%29%2F%284%29 ALIGN=MIDDLE  ALT=\"%283x%28900-3x%5E2%29%29%2F%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "V = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%282700x+-+9x%5E3%29%29%2F4 ALIGN=MIDDLE  ALT=\"%28%282700x+-+9x%5E3%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or 675x - 2.25x^3<BR>\n" );
document.write( "Plot this equation, volume vertical, width horizontal<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-6%2C+20%2C+-1000%2C+5000%2C+675x-2.25x%5E3%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-6%2C+20%2C+-1000%2C+5000%2C+675x-2.25x%5E3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "We can see max volume occurs when x=10 which is the width<BR>\n" );
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document.write( "Find the dimensions<BR>\n" );
document.write( "L = 3(10)<BR>\n" );
document.write( "L = 30 cm<BR>\n" );
document.write( "w = 10 cm<BR>\n" );
document.write( "h = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28900-3%2810%5E2%29%29%29%2F%284%2A10%29 ALIGN=MIDDLE  ALT=\"%28%28900-3%2810%5E2%29%29%29%2F%284%2A10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "h = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28900-300%29%2F40 ALIGN=MIDDLE  ALT=\"%28900-300%29%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "h = 15 cm is the height<BR>\n" );
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document.write( "30 * 10 * 15 = 4500 cu/cm is max volume<BR>\n" );
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document.write( "Check this by finding the surface area with these dimensions<BR>\n" );
document.write( "2(30*10) + 2(30*15) + 2(10*15) =<BR>\n" );
document.write( "600 + 900 + 300 = 1800 sq cm \r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );