document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=310349>Question 310349</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The base of a rectangle is on the x-axis and its two upper vertices are on the parabola y=16-x^2. of all such rectangles, what are the dimensions of the one with greatest area? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #221898 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=221898\">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( "Because of the fact that the parabola is symmetric to the y-axis, the rectangle must also be symmetric to the y-axis.  That means that the two lower vertices are <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(-x,0\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x,0\right)\">.  The measure of the base of the rectangle is therefore <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2x\">.  The upper vertices, being points on the parabola are:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(-x,16\ -\ x^2\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x,16\ -\ x^2\right)\">.  Therefore, the measure of the height of the rectangle is simply <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 16\ -\ x^2\">.  Therefore the area of the rectangle is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(x)\ =\ \left(2x\right)\left(16\ -\ x^2\right)\ =\ 32x\ -\ 2x^3\">\r<BR>\n" );
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document.write( "Such a function has local extremes at the points where the first derivative is zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A'(x)\ =\ 32\ -\ 6x^2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 32\ -\ 6x^2\ =\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \pm\sqrt{\frac{32}{6}}\ =\ \pm\sqrt{\frac{16}{3}}\ =\ \pm\frac{4\sqrt{3}}{3}\">\r<BR>\n" );
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document.write( "Discard the negative root since we need a positive measure of length\r<BR>\n" );
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document.write( "We are looking for a maximum, so we want to see if the value of the second derivative is negative at the extreme point.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A{''}(\frac{4\sqrt{3}}{3})\ =\ -12\left(\frac{4\sqrt{3}}{3}\right)\ <\ 0\">\r<BR>\n" );
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document.write( "Which it is...\r<BR>\n" );
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document.write( "So, the horizontal dimension of the largest area rectangle is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\ \cdot\ \frac{4\sqrt{3}}{3}\ =\ \frac{8\sqrt{3}}{3}\">\r<BR>\n" );
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document.write( "And the vertical dimension is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 16\ -\ \left(\frac{4\sqrt{3}}{3}\right)^2\ =\ \frac{32}{3}\"> \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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