document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1103715>Question 1103715</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A piece of wire 64 cm long is cut into two pieces with different lengths. The two pieces are formed to TWO squares.<BR>\n" );
document.write( " (a) Show that the combined area of the two squares A is given by A=2x^2-30x+256<BR>\n" );
document.write( " (b) What is the minimum combined area of the two squares? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #718437 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=718437\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A piece of wire 64 cm long is cut into two pieces with different lengths. The two pieces are formed to TWO squares.<BR>\n" );
document.write( " (a) Show that the combined area of the two squares A is given by A=2x^2-30x+256<BR>\n" );
document.write( " (b) What is the minimum combined area of the two squares?<BR>\n" );
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document.write( "Let x = the length of the first piece<BR>\n" );
document.write( "the  64-x = the length of the 2nd piece\r<BR>\n" );
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document.write( "Area of the first square is   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x%2F4%29%5E2+ ALIGN=MIDDLE  ALT=\"+%28x%2F4%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Aree of the 2nd square is   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28%2864-x%29%2F4%29%5E2+ ALIGN=MIDDLE  ALT=\"+%28%2864-x%29%2F4%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Combined area = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x%5E2%2F16%29+%2B+%28%284096-128x%2Bx%5E2%29%2F16%29+ ALIGN=MIDDLE  ALT=\"+%28x%5E2%2F16%29+%2B+%28%284096-128x%2Bx%5E2%29%2F16%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This simplifies to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28matrix%281%2C3%2C+%22+%22%2C+%281%2F8%29%28x%5E2-64x%2B2048%29%2C%22+%22%29%29+ ALIGN=MIDDLE  ALT=\"+highlight%28matrix%281%2C3%2C+%22+%22%2C+%281%2F8%29%28x%5E2-64x%2B2048%29%2C%22+%22%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice that this disagrees with the value of A in the problem statement.  I checked my work but I've been known to make errors.<BR>\n" );
document.write( "—————————<BR>\n" );
document.write( "To find the minimum, take the derivative.  I will use my equation (the process is the same even if the correct equation is different):<BR>\n" );
document.write( "    <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+%281%2F8%29%28x%5E2-64x%2B2048%29+ ALIGN=MIDDLE  ALT=\"+A+=+%281%2F8%29%28x%5E2-64x%2B2048%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Take the derivative of A with respect to x:<BR>\n" );
document.write( "   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+dA%2Fdx+=+%281%2F8%29%282x+-+64%29+ ALIGN=MIDDLE  ALT=\"+dA%2Fdx+=+%281%2F8%29%282x+-+64%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Set it to zero:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+++%281%2F8%29%282x-64%29+=+0++ ALIGN=MIDDLE  ALT=\"+++%281%2F8%29%282x-64%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=x+=+32cm ALIGN=MIDDLE  ALT=\"x+=+32cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( " Min Area = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%2832%2F4%29%5E2+%2B+%2832%2F4%29%5E2+=+8%5E2+%2B+8%5E2+=+highlight%28128cm%5E2%29+ ALIGN=MIDDLE  ALT=\"+%2832%2F4%29%5E2+%2B+%2832%2F4%29%5E2+=+8%5E2+%2B+8%5E2+=+highlight%28128cm%5E2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "This is a local minimum of A because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+d%5E2A%2Fdx%5E2+=+%281%2F8%29%282%29+ ALIGN=MIDDLE  ALT=\"+d%5E2A%2Fdx%5E2+=+%281%2F8%29%282%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > —> concave up.  Normally, you should also check the endpoints (here x=0, x=64) but due to the quadratic form, we know the local minimum is a global minimum on the interval.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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