document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=168001>Question 168001</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a wire 10 cm long is cut into two pieces, one of length x and the other of 10 - x. each piece is bent into the shape of a square<BR>\n" );
document.write( "(a) find a function that models the total area enclosed by the two squares.<BR>\n" );
document.write( "(b) Find the Value of x that minimizes the total 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 #123862 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=123862\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> area of a square is s^2 where s is the length of each side.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the wire is cut into 2 lengths and forms a square with each.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the wire of x length forms 4 sides with the length of each side equal to x/4.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the wire of (10-x) length forms 4 sides with the length of each side equal to (10-x)/4.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "area of the square made by the wire of x length 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( "this equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2F+4%5E2 ALIGN=MIDDLE  ALT=\"x%5E2+%2F+4%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F16 ALIGN=MIDDLE  ALT=\"x%5E2%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "area of the square made by the wire of (10-x) length is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%2810-x%29%2F4%29%5E2 ALIGN=MIDDLE  ALT=\"%28%2810-x%29%2F4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "this equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810-x%29%5E2+%2F+4%5E2 ALIGN=MIDDLE  ALT=\"%2810-x%29%5E2+%2F+4%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810-x%29%5E2+%2F+16 ALIGN=MIDDLE  ALT=\"%2810-x%29%5E2+%2F+16\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the total area enclosed by these squares is given by the equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+x%5E2%2F16+%2B+%2810-x%29%5E2%2F16 ALIGN=MIDDLE  ALT=\"f%28x%29+=+x%5E2%2F16+%2B+%2810-x%29%5E2%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "since the denominator is equal, this equation becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%28x%5E2+%2B+%2810-x%29%5E2%29%2F16 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%28x%5E2+%2B+%2810-x%29%5E2%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since x^2 is positive, this is a quadratic equation that opens upward and points downward.<BR>\n" );
document.write( "the point where the graph turns is a minimum point.<BR>\n" );
document.write( "that point can be found as follows:<BR>\n" );
document.write( "take the given formula and multiply it out to get a quadratic equation in standard form.<BR>\n" );
document.write( "that standard form is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Ax%5E2+-+20%2Ax+%2B+100%29%2F16 ALIGN=MIDDLE  ALT=\"%282%2Ax%5E2+-+20%2Ax+%2B+100%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "that becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2F8%29+-+%2810%2Ax%2F8%29+%2B+%2850%2F8%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2F8%29+-+%2810%2Ax%2F8%29+%2B+%2850%2F8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "in that equation, the a coefficient is 1/8, and the b coefficient is -10/8.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a ALIGN=MIDDLE  ALT=\"-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives the minimum/ maximum point of a quadratic equation.<BR>\n" );
document.write( "substituting, that formula becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-+%28-10%2F8%29+%2F+%282%2F8%29+=+5 ALIGN=MIDDLE  ALT=\"-+%28-10%2F8%29+%2F+%282%2F8%29+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "when x = 5, y = 3.125<BR>\n" );
document.write( "that's the minimum value of the equation.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "graph of the resulting equation looks like this showing that when x = 5, y = 3.125 and the value of the equation of the squares is at a minimum.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28800%2C800%2C-1%2C9%2C-10%2C10%2C%28x%5E2+%2B+%2810-x%29%5E2%29%2F16%29 ALIGN=MIDDLE  ALT=\"graph%28800%2C800%2C-1%2C9%2C-10%2C10%2C%28x%5E2+%2B+%2810-x%29%5E2%29%2F16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );