document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=549580>Question 549580</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A piece of wire 16 cm long is cut into two pieces, with the first piece having length x. The first piece is formed into a rectangle in which the length is twice the width. The second piece of wire is also formed into a rectangle, but with the length three times the width. For what value of x is the total area of the two rectangles a minimum? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #357888 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=357888\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> is the perimeter of the first rectangle where the length is twice the width, we can say:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 6w\small{_1}\LARGE\ =\ x\">\r<BR>\n" );
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document.write( "From which we can derive:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ w\small{_1}\LARGE\ =\ \frac{x}{6}\">\r<BR>\n" );
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document.write( "Likewise, since the perimeter of the other rectangle is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 16\ -\ x\"> and the length of the second rectangle is 3 times the width, we can say:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 8w\small{_2}\LARGE\ =\ 16\ -\ x\">\r<BR>\n" );
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document.write( "From which we can derive:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ w\small{_2}\LARGE\ =\ \frac{16\ -\ x}{8} \">\r<BR>\n" );
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document.write( "The area of the first rectangle in terms of its width:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2w\small{_1}\LARGE\ \cdot\ w\small{_1}\LARGE\ =\ 2w\small{_1}\LARGE^2\">\r<BR>\n" );
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document.write( "And the area of the second rectangle in terms of its width is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3w\small{_2}\LARGE\ \cdot\ w\small{_2}\LARGE\ =\ 3w\small{_2}\LARGE^2\">\r<BR>\n" );
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document.write( "Then the total area is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2w\small{_1}\LARGE^2\ +\ 3w\small{_2}\LARGE\,^2\">\r<BR>\n" );
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document.write( "Substituting the earlier derived expressions for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large w\small{_1}\LARGE\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large w\small{_2}\LARGE\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(x)\ =\ 2\left(\frac{x}{6}\right)^2\ +\ 3\left(\frac{16\ -\ x}{8}\right)^2\">\r<BR>\n" );
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document.write( "If you expand, collect terms, and find a common denominator, you will have a function that is a quadratic trinomial with a positive lead coefficient, i.e. the equation of a convex up parabola.\r<BR>\n" );
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document.write( "If this is an algebra problem, use the formula for the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of the vertex of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x)\ =\ ax^2\ +\ bx\ +\ c\">, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x_v\ =\ \frac{-b}{2a}\"> to get your answer.\r<BR>\n" );
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document.write( "If this is a calculus problem, take the first derivative and set it equal to zero.  Solve the resulting equation.  Verify that the second derivative evaluates to a positive number at this value of the independent variable.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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