document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=761289>Question 761289</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am stuck with where to begin... I think I need the perimeter of rectangle, area of rectangle and half circumference of circle...<BR>\n" );
document.write( "A norman window is a rectangle with a semicircle on top. Design a window requiring 24 ft of trim on outer edges.  What dimensions will allow the maximum amount of light to enter a house?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #463167 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=463167\">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( "First thing is to make the rather reasonable assumption that the maximum amount of light would be admitted by the maximum area window.\r<BR>\n" );
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document.write( "The perimeter of the overall window is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \pi r\ +\ 2r\ +\ 2h\ =\ 24\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE h\"> is the height of the rectangular part of the window and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE r\"> is the radius of the semi-circular piece.\r<BR>\n" );
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document.write( "From this relationship we can determine that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE h\ =\ 12\ -\ \left(1\ -\frac{\pi}{2}\right)r\"> \r<BR>\n" );
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document.write( "The area of the rectangular portion of the window is then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A_r\ =\  24r\ -\ \left(2\ +\ \pi\right)r^2\">\r<BR>\n" );
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document.write( "The area of the semi-circular portion of the window is  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A_s\ =\ \frac{\pi}{2}r^2\">\r<BR>\n" );
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document.write( "Add 'em up:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A_w(r)\ =\ 24r\ -\ \left(2\ +\ \pi\right)r^2\ +\ \frac{\pi}{2}r^2\">\r<BR>\n" );
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document.write( "Take the first derivative: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{d}{dr}A_w(r)\ =\ 24\ -\ \left(4\ +\ 2\pi\right)r\ +\ \pi r\ =\ 24\ -\ (4\ +\ \pi)r\">\r<BR>\n" );
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document.write( "Set the derivative equal to zero: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 24\ -\ (4\ +\ \pi)r\ =\ 0\">\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE r\ =\ \frac{24}{4\ +\ \pi}\">\r<BR>\n" );
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document.write( "Take the second derivative:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{d^2}{dr^2}A_w(r)\ =\ -4\ -\ \pi\"> which is negative for all values of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE r\">.  So the function is concave down in the region of the extremum and therefore the value of the function at that point is a maximum.\r<BR>\n" );
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document.write( "So the rectangular part of the window must have a width of twice the radius of the semi-circle, or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE w\ = \frac{48}{4\ +\ \pi}\">, and the height must be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE h\ =\ 12\ -\ \left(1\ -\frac{\pi}{2}\right)\left(\frac{24}{4\ +\ \pi}\right\) =\ \frac{24}{4\ +\ \pi}\">\r<BR>\n" );
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document.write( "I'll leave the verification of that simplification to you.\r<BR>\n" );
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document.write( "That's all you need to answer your question.\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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