document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=403095>Question 403095</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A norman window has the shape of a rectangle surmounted by a semicircle of a diameter equal to the width of the rectangle. If the permimeter of the window is 20 feet, what dimensions will admit the most light (maximize the area)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #285137 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=285137\">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( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> represent the measure of the vertical dimension of the rectangular portion of the window.  Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> represent the horizontal dimension of the window.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> is also the diameter of the semi-circular part of the window.\r<BR>\n" );
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document.write( "The perimeter of the window is then:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P(x,y)\ =\ 2y\ +\ x\ +\ \frac{\pi x}{2}\">\r<BR>\n" );
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document.write( "But we are given that the perimeter is 20 feet, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2y\ +\ x\ +\ \frac{\pi x}{2}\ =\ 20\">\r<BR>\n" );
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document.write( "Solve for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 10\ -\ \frac{(2\ +\ \pi)x}{4} \">\r<BR>\n" );
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document.write( "The area of the window is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(x,y)\ =\ xy\ +\ \frac{\pi{\frac{x}{2}}^2}{2}\">\r<BR>\n" );
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document.write( "Substitute to create a function for area in terms of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(x)\ =\ 10x\ -\ \frac{(2\ +\ \pi)x^2}{4}\ +\ \frac{\pi{\frac{x}{2}}^2}{2}\">\r<BR>\n" );
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document.write( "Simplify:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(x)\ =\ 10x\ -\ \frac{(4\ +\ \pi)}{8}x^2\">\r<BR>\n" );
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document.write( "Two ways to go from here:\r<BR>\n" );
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document.write( "<u><b>Algebra Method</b></u>\r<BR>\n" );
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document.write( "This is a quadratic with a negative lead coefficient, hence the graph is a parabola that opens downward.  That means the vertex is a maximum.  Recall that the vertex of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \rho(x)\ =\ ax^2\ +\ bx\ +\ c\"> is at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x_v = \frac{-b}{2a}\">.\r<BR>\n" );
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document.write( "For this problem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_{max} = \frac{-10}{-\frac{(4\ +\ \pi)}{4}}\ =\ \frac{40}{4\ +\ \pi}\">\r<BR>\n" );
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document.write( "and a little arithmetic gets us to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y_{max}\ =\ \frac{10(2\ +\ \pi)}{4\ +\ \pi}\">\r<BR>\n" );
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document.write( "And the radius of the semi-circle is then:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{x_{max}}{2} = \frac{20}{4\ +\ \pi}\">\r<BR>\n" );
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document.write( "<u><b>Calculus Method</b></u>\r<BR>\n" );
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document.write( "This is a 2nd degree polynomial equation, hence continuous and twice differentiable over the real numbers.  Therefore there will be a local extremum at any point where the first derivative is equal to zero and that point will be a maximum if the 2nd derivative is negative at that point.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dA}{dx}\ =\ 10\ -\ \frac{(4\ +\ \pi)}{4}x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 10\ -\ \frac{(4\ +\ \pi)}{4}x\ =\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = \frac{40}{4\ +\ \pi}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2A}{dx^2}\ =\ -\ \frac{(4\ +\ \pi)}{4}\ <\ 0\">\r<BR>\n" );
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document.write( "Hence <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = \frac{40}{4\ +\ \pi}\">\r<BR>\n" );
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document.write( "is the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of the maximum point.\r<BR>\n" );
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document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> dimension and the semi-circle radius are calculated as above.\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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