document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=199484>Question 199484</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Problems like these always make sure I come back to this website for help =( ; Find the dimension of a rectangle “a” with the greatest area whose perimeter is 30 ft.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #149891 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=149891\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "The perimeter of a rectangle is given by the formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P = 2l + 2w\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large l\"> is the measure of the length of the rectangle and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large w\"> is the measure of the width.\r<BR>\n" );
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document.write( "Solving for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large l\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ l = \frac{P - 2w}{2}\">\r<BR>\n" );
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document.write( "The area of a rectangle is given by the formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A = lw\">\r<BR>\n" );
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document.write( "Substituting the expression for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large l\"> derived earlier, you can write a function for the area of the rectangle in terms of the width where the perimeter is a constant:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(x) = \left(\frac{P - 2w}{2}\right)w = - w^2 + \frac{1}{2}Pw\">\r<BR>\n" );
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document.write( "This is a quadratic function whose graph is a parabola.  Since the lead coefficient is <0, the parabola opens downward and the vertex of the parabola represents a maximum.  Since the independent variable is the width and the value of the function is the area, the coordinates of the vertex will tell us the width in terms of the perimeter that gives us the maximum area, and the value of that maximum area, again in terms of the perimeter.\r<BR>\n" );
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document.write( "A parabola represented by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x) = ax^2 + bx + c\"> has a vertex at the point:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)\">\r<BR>\n" );
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document.write( "For the area function derived above, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a = -1\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b = \frac{P}{2}\">, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{-\frac{P}{2}}{-2} = \frac{P}{4}\">\r<BR>\n" );
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document.write( "Hence, the maximum area is obtained when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large w = \frac{P}{4}\">, which means that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2w = \frac{P}{2}\"> and therefore <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2l = \frac{P}{2}\"> also.  Therefore, the maximum area for a given perimeter is obtained when the rectangle is actually a square with side measure one-fourth of the perimeter.\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" );
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