document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=971976>Question 971976</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the area of the largest trapezoid that can be inscribed a semicircle of diameter \"d\" in terms of \"d\"?. what is the length of the bottom base \"x\" of the same trapezoid ?? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #594442 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=594442\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In the American definition, a trapezoid has two parallel sides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C200%2C-1.2%2C1.2%2C-0.3%2C1.3%2C%0D%0Ared%28arc%280%2C0%2C2%2C2%2C180%2C360%29%29%2C%0D%0Aline%28-1%2C0%2C-0.5%2C0.866%29%2Cline%281%2C0%2C0.5%2C0.866%29%2C%0D%0Aline%28-1%2C0%2C1%2C0%29%2Cline%28-0.5%2C0.866%2C0.5%2C0.866%29%2C%0D%0Aarrow%28-0.5%2C0.866%2C0%2C0.866%29%2Carrow%28-0.5%2C0%2C0%2C0%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "I am assuming that the drawing above shows what was meant: a trapezoid inscribed in a semicircle.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Intuitively, we could assume that the largest polygon that can be inscribed in a circle is the regular polygon of its kind,<BR>\n" );
document.write( "so that the largest inscribed triangle would be an equilateral triangle,<BR>\n" );
document.write( "the largest quadrilateral would be a square,<BR>\n" );
document.write( "the largest pentagon a regular pentagon,<BR>\n" );
document.write( "the largest hexagon a regular hexagon, and so on.<BR>\n" );
document.write( "If that is so, the largest trapezoid inscribed in a semicircle may be half of a hexagon.<BR>\n" );
document.write( "That trapezoid would have base angles measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60%5Eo ALIGN=MIDDLE  ALT=\"60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "and could be thought of as made of three equilateral triangles: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C200%2C-1.2%2C1.2%2C-0.3%2C1.3%2C%0D%0Ared%28arc%280%2C0%2C2%2C2%2C180%2C360%29%29%2C%0D%0Atriangle%28-1%2C0%2C-0.5%2C0.866%2C0%2C0%29%2Ctriangle%281%2C0%2C0.5%2C0.866%2C0%2C0%29%2C%0D%0Atriangle%28-0.5%2C0.866%2C0.5%2C0.866%2C0%2C0%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In a semicircle of radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=d%2F2 ALIGN=MIDDLE  ALT=\"r=d%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,the area of that trapezoid is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=3%2A%281%2F2%29r%2Ar%2Asin%2860%5Eo%29=3%2A%281%2F2%29%2Ar%5E2%2A%28sqrt%283%29%2F2%29=3%2Ar%5E2sqrt%283%29%2F4=3%2A%28r%2F2%29%5E2%2Asqrt%283%29=3sqrt%283%29%28d%2F4%29%5E2=3sqrt%283%29d%5E2%2F16 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "There may be a very simple proof supporting that maximum  area value,<BR>\n" );
document.write( "but I was taught a lot of math past the fifth grade, and that spoiled me forever,<BR>\n" );
document.write( "so I can only offer a complicated proof or two.<BR>\n" );
document.write( "I just cannot think of a proof that does not involve calculus.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C200%2C-1.2%2C1.2%2C-0.4%2C1.2%2C%0D%0Ared%28arc%280%2C0%2C2%2C2%2C180%2C360%29%29%2Clocate%280.45%2C0.45%2Cr%29%2C%0D%0Atriangle%28-1%2C0%2C-0.707%2C0.707%2C0%2C0%29%2Cgreen%28triangle%281%2C0%2C0.707%2C0.707%2C0.707%2C0%29%29%2C%0D%0Agreen%28rectangle%280.707%2C0%2C0.66%2C0.05%29%29%2Ctriangle%281%2C0%2C0.707%2C0.707%2C0%2C0%29%2C%0D%0Atriangle%28-0.707%2C0.707%2C0.707%2C0.707%2C0%2C0%29%2Clocate%280.15%2C0.15%2Cred%28alpha%29%29%2C%0D%0Alocate%28-0.25%2C0.15%2Cred%28alpha%29%29%2Cred%28arc%280%2C0%2C0.6%2C0.6%2C180%2C225%29%29%2C%0D%0Ared%28arc%280%2C0%2C0.6%2C0.6%2C-45%2C0%29%29%2Cred%28arc%280%2C0%2C0.5%2C0.5%2C-135%2C-45%29%29%2C%0D%0Alocate%28-0.22%2C0.4%2Cred%28180%5Eo-2alpha%29%29%2Clocate%280.72%2C0.37%2Cgreen%28h%29%29%2C%0D%0Agreen%28arrow%28-0.3%2C-0.1%2C-1%2C-0.1%29%29%2Cgreen%28arrow%28-0.7%2C-0.1%2C0%2C-0.1%29%29%2C%0D%0Alocate%28-0.7%2C-0.1%2Cgreen%28r=d%2F2%29%29%2Clocate%280.3%2C-0.1%2Cgreen%28x%29%29%2C%0D%0Agreen%28arrow%280%2C-0.1%2C0.707%2C-0.1%29%29%2Cgreen%28arrow%280.707%2C-0.1%2C0%2C-0.1%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28x%29 ALIGN=MIDDLE  ALT=\"green%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=sqrt%28r%5E2-x%5E2%29 ALIGN=MIDDLE  ALT=\"h=sqrt%28r%5E2-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=A=%282r%2B2x%29sqrt%28r%5E2-x%5E2%29%2F2=%28r%2Bx%29sqrt%28r%5E2-x%5E2%29 ALIGN=MIDDLE  ALT=\"area=A=%282r%2B2x%29sqrt%28r%5E2-x%5E2%29%2F2=%28r%2Bx%29sqrt%28r%5E2-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdx=1%2Asqrt%28r%5E2-x%5E2%29%2B%28r%2Bx%29%2A%281%2F2sqrt%28r%5E2-x%5E2%29%29%2A%28-2x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdx=sqrt%28r%5E2-x%5E2%29%2B%28r%2Bx%29%2Ax%2Fsqrt%28r%5E2-x%5E2%29%29 ALIGN=MIDDLE  ALT=\"dA%2Fdx=sqrt%28r%5E2-x%5E2%29%2B%28r%2Bx%29%2Ax%2Fsqrt%28r%5E2-x%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdx=%28r%5E2-x%5E2%29%2Fsqrt%28r%5E2-x%5E2%29-%28rx%2Bx%5E2%29%2Fsqrt%28r%5E2-x%5E2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdx=%28r%5E2-x%5E2-rx-x%5E2%29%2Fsqrt%28r%5E2-x%5E2%29%29 ALIGN=MIDDLE  ALT=\"dA%2Fdx=%28r%5E2-x%5E2-rx-x%5E2%29%2Fsqrt%28r%5E2-x%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdx=%28-2x%5E2-rx%2Br%5E2%29%2Fsqrt%28r%5E2-x%5E2%29%29 ALIGN=MIDDLE  ALT=\"dA%2Fdx=%28-2x%5E2-rx%2Br%5E2%29%2Fsqrt%28r%5E2-x%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdx%3E0 ALIGN=MIDDLE  ALT=\"dA%2Fdx%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > between both roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E2-rx%2Br%5E2=0 ALIGN=MIDDLE  ALT=\"-2x%5E2-rx%2Br%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2Brx-r%5E2=0 ALIGN=MIDDLE  ALT=\"2x%5E2%2Brx-r%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solving that quadratic equation we realize that the solutions are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-r ALIGN=MIDDLE  ALT=\"x=-r\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which, making <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C0 ALIGN=MIDDLE  ALT=\"x%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not make sense, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=r%2F2=d%2F4 ALIGN=MIDDLE  ALT=\"x=r%2F2=d%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which makes sense and makes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdx=0 ALIGN=MIDDLE  ALT=\"dA%2Fdx=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "marking the maximum for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28r%2Bx%29sqrt%28r%5E2-x%5E2%29 ALIGN=MIDDLE  ALT=\"A=%28r%2Bx%29sqrt%28r%5E2-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, that maximum is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=%28r%2Br%2F2%29sqrt%28r%5E2-%28r%2F2%29%5E2%29 ALIGN=MIDDLE  ALT=\"area%5Bmax%5D=%28r%2Br%2F2%29sqrt%28r%5E2-%28r%2F2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=%283r%2F2%29sqrt%28r%5E2-r%5E2%2F4%29 ALIGN=MIDDLE  ALT=\"area%5Bmax%5D=%283r%2F2%29sqrt%28r%5E2-r%5E2%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=%283r%2F2%29sqrt%283r%5E2%2F4%29 ALIGN=MIDDLE  ALT=\"area%5Bmax%5D=%283r%2F2%29sqrt%283r%5E2%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=%283r%2F2%29sqrt%283%29%28r%2F2%29%29 ALIGN=MIDDLE  ALT=\"area%5Bmax%5D=%283r%2F2%29sqrt%283%29%28r%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=3sqrt%283%29%28r%2F2%29%5E2%29 ALIGN=MIDDLE  ALT=\"area%5Bmax%5D=3sqrt%283%29%28r%2F2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=3sqrt%283%29%28r%2F4%29%5E2%29 ALIGN=MIDDLE  ALT=\"area%5Bmax%5D=3sqrt%283%29%28r%2F4%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=highlight%283sqrt%283%29d%5E2%2F16%29 ALIGN=MIDDLE  ALT=\"area%5Bmax%5D=highlight%283sqrt%283%29d%5E2%2F16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28alpha%29 ALIGN=MIDDLE  ALT=\"red%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "The trapezoid is made of three isosceles triangles.<BR>\n" );
document.write( "They have legs of length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and vertex angles measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28alpha%29 ALIGN=MIDDLE  ALT=\"red%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28180%5Eo-2alpha%29 ALIGN=MIDDLE  ALT=\"red%28180%5Eo-2alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28alpha%29 ALIGN=MIDDLE  ALT=\"red%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The area of the trapezoid is the sum of the triangles' areas, so it is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=A=r%5E2sin%28alpha%29%2F2%2Br%5E2sin%28180%5Eo-2alpha%29%2F2%2Br%5E2sin%28alpha%29%2F2 ALIGN=MIDDLE  ALT=\"area=A=r%5E2sin%28alpha%29%2F2%2Br%5E2sin%28180%5Eo-2alpha%29%2F2%2Br%5E2sin%28alpha%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=r%5E2sin%28alpha%29%2B%28r%5E2%2F2%29sin%28180%5Eo-2alpha%29 ALIGN=MIDDLE  ALT=\"A=r%5E2sin%28alpha%29%2B%28r%5E2%2F2%29sin%28180%5Eo-2alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fd%28alpha%29=r%5E2%28-cos%28alpha%29%29%2B%28r%5E2%2F2%29%28-cos%28180%5Eo-2alpha%29%29%28-1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fd%28alpha%29=-r%5E2cos%28alpha%29%2B%28r%5E2%2F2%29cos%28180%5Eo-2alpha%29 ALIGN=MIDDLE  ALT=\"dA%2Fd%28alpha%29=-r%5E2cos%28alpha%29%2B%28r%5E2%2F2%29cos%28180%5Eo-2alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fd%28alpha%29=-r%5E2cos%28alpha%29%2B%28r%5E2%2F2%29cos%28180%5Eo%29cos%282alpha%29%2Bsin%28180%5Eo%29sin%282alpha%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fd%28alpha%29=-r%5E2cos%28alpha%29%2B%28r%5E2%2F2%29%28-1%29cos%282alpha%29%2B0%2Asin%282alpha%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fd%28alpha%29=-r%5E2cos%28alpha%29-%28r%5E2%2F2%29cos%282alpha%29 ALIGN=MIDDLE  ALT=\"dA%2Fd%28alpha%29=-r%5E2cos%28alpha%29-%28r%5E2%2F2%29cos%282alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fd%28alpha%29=-r%5E2cos%28alpha%29-%28r%5E2%2F2%29%282cos%5E2%28alpha%29-1%29 ALIGN=MIDDLE  ALT=\"dA%2Fd%28alpha%29=-r%5E2cos%28alpha%29-%28r%5E2%2F2%29%282cos%5E2%28alpha%29-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fd%28alpha%29=-%28r%5E2%2F2%29%282cos%28alpha%29%2B2cos%5E2%28alpha%29-1%29 ALIGN=MIDDLE  ALT=\"dA%2Fd%28alpha%29=-%28r%5E2%2F2%29%282cos%28alpha%29%2B2cos%5E2%28alpha%29-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fd%28alpha%29=0 ALIGN=MIDDLE  ALT=\"dA%2Fd%28alpha%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2cos%28alpha%29%2B2cos%5E2%28alpha%29-1=0 ALIGN=MIDDLE  ALT=\"2cos%28alpha%29%2B2cos%5E2%28alpha%29-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fd%28alpha%29%3E=0 ALIGN=MIDDLE  ALT=\"dA%2Fd%28alpha%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > between both roots,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bmax%5D ALIGN=MIDDLE  ALT=\"A%5Bmax%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > happens for the greatest root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2cos%28alpha%29%2B2cos%5E2%28alpha%29-1=0 ALIGN=MIDDLE  ALT=\"2cos%28alpha%29%2B2cos%5E2%28alpha%29-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Making <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%29=u ALIGN=MIDDLE  ALT=\"cos%28alpha%29=u\"   DISABLED_style= \"max-width:90%; height:auto;\" > for short, the equation is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2u%5E2%2B2u-1=0 ALIGN=MIDDLE  ALT=\"2u%5E2%2B2u-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the roots are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+%28-1+%2B-+sqrt%281%5E2-4%2A2%2A%28-1%29%29%29%2F%282%2A2%29+ ALIGN=MIDDLE  ALT=\"u+=+%28-1+%2B-+sqrt%281%5E2-4%2A2%2A%28-1%29%29%29%2F%282%2A2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=%28-1+%2B-+sqrt%281%2B8%29%29%2F4=%28-1+%2B-+sqrt%289%29%29%2F4=%28-1+%2B-+3%29%2F4 ALIGN=MIDDLE  ALT=\"u+=%28-1+%2B-+sqrt%281%2B8%29%29%2F4=%28-1+%2B-+sqrt%289%29%29%2F4=%28-1+%2B-+3%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5B1%5D=%28-1-3%29%2F4=%28-4%29%2F4=-1%3C%28-1%2B3%29%2F4=2%2F4=1%2F2 ALIGN=MIDDLE  ALT=\"u%5B1%5D=%28-1-3%29%2F4=%28-4%29%2F4=-1%3C%28-1%2B3%29%2F4=2%2F4=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area ALIGN=MIDDLE  ALT=\"area\"   DISABLED_style= \"max-width:90%; height:auto;\" > increases with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%29 ALIGN=MIDDLE  ALT=\"cos%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%29=0 ALIGN=MIDDLE  ALT=\"cos%28alpha%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha=90%5Eo ALIGN=MIDDLE  ALT=\"alpha=90%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%29=1%2F2 ALIGN=MIDDLE  ALT=\"cos%28alpha%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha=60%5Eo ALIGN=MIDDLE  ALT=\"alpha=60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the maximum is at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%29=1%2F2 ALIGN=MIDDLE  ALT=\"cos%28alpha%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha=60%5Eo ALIGN=MIDDLE  ALT=\"alpha=60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > , where<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=r%5E2sin%28alpha%29%2Br%5E2sin%28180%5Eo-2alpha%29%2F2 ALIGN=MIDDLE  ALT=\"area%5Bmax%5D=r%5E2sin%28alpha%29%2Br%5E2sin%28180%5Eo-2alpha%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=r%5E2sin%2860%5Eo%29%2Br%5E2sin%28180%5Eo-2%2A60%5Eo%29%2F2 ALIGN=MIDDLE  ALT=\"area%5Bmax%5D=r%5E2sin%2860%5Eo%29%2Br%5E2sin%28180%5Eo-2%2A60%5Eo%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=r%5E2sin%2860%5Eo%29%2Br%5E2sin%2860%5Eo%29%2F2 ALIGN=MIDDLE  ALT=\"area%5Bmax%5D=r%5E2sin%2860%5Eo%29%2Br%5E2sin%2860%5Eo%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=%283r%5E2%2F2%29sin%2860%5Eo%29 ALIGN=MIDDLE  ALT=\"area%5Bmax%5D=%283r%5E2%2F2%29sin%2860%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=%283r%5E2%2F2%29%28sqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"area%5Bmax%5D=%283r%5E2%2F2%29%28sqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=%283%2F4%29r%5E2sqrt%283%29%7D%7D+%2C+and+since+%7B%7B%7Br=d%2F2 ALIGN=MIDDLE  ALT=\"area%5Bmax%5D=%283%2F4%29r%5E2sqrt%283%29%7D%7D+%2C+and+since+%7B%7B%7Br=d%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%5Bmax%5D=%283%2F4%29%28d%2F2%29%5E2sqrt%283%29=%283%2F4%29%28d%5E2%2F4%29sqrt%283%29=highlight%283sqrt%283%29d%5E2%2F16%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );