document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1060565>Question 1060565</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A trough for feeding cattle is 16 feet long and it's cross sections are isosceles triangles with two equal sides being 18 inches. The angle between the two equal sides is theta. <BR>\n" );
document.write( "A) express the volume of the trough as a function of theta/2<BR>\n" );
document.write( "B) express the volume of the trough as a function of theta and determine the value of theta so the volume is maximum.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #675505 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=675505\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If the isosceles triangle is cut in half such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+theta+ ALIGN=MIDDLE  ALT=\"+theta+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is bisected, then<BR>\n" );
document.write( "the side adjacent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+theta%2F2+ ALIGN=MIDDLE  ALT=\"+theta%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+18%2Acos%28theta%2F2%29+ ALIGN=MIDDLE  ALT=\"+18%2Acos%28theta%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the side opposite<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta%2F2 ALIGN=MIDDLE  ALT=\"theta%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+18%2Asin%28theta%2F2%29+ ALIGN=MIDDLE  ALT=\"+18%2Asin%28theta%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_half+=+%281%2F2%29%2A%2818%2Asin%28theta%2F2%29%29%2A%2818%2Acos%28theta%2F2%29%29+ ALIGN=MIDDLE  ALT=\"+Area_half+=+%281%2F2%29%2A%2818%2Asin%28theta%2F2%29%29%2A%2818%2Acos%28theta%2F2%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "--<BR>\n" );
document.write( "The full area of the isosceles triangle is twice this:\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Area+=+18%2Asin%28theta%2F2%29+%2A+18%2Acos%28theta%2F2%29+ ALIGN=MIDDLE  ALT=\"+Area+=+18%2Asin%28theta%2F2%29+%2A+18%2Acos%28theta%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Area+=+324%2Asin%28theta%2F2%29%2Acos%28theta%2F2%29+ ALIGN=MIDDLE  ALT=\"+Area+=+324%2Asin%28theta%2F2%29%2Acos%28theta%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >    (this is sq in)\r<BR>\n" );
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document.write( "Adjust the formula to give us sq ft (sq ft = sq in / 144):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Area+=+%28324%2F144%29%2Asin%28theta%2F2%29%2Acos%28theta%2F2%29+ ALIGN=MIDDLE  ALT=\"+Area+=+%28324%2F144%29%2Asin%28theta%2F2%29%2Acos%28theta%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   sq ft<BR>\n" );
document.write( "—\r<BR>\n" );
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document.write( "The trough volume is just 16 ft times this area:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Volume+=+16%2A%28324%2F144%29%2Asin%28theta%2F2%29%2Acos%28theta%2F2%29+ ALIGN=MIDDLE  ALT=\"+Volume+=+16%2A%28324%2F144%29%2Asin%28theta%2F2%29%2Acos%28theta%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ft%5E3+ ALIGN=MIDDLE  ALT=\"+ft%5E3+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "—\r<BR>\n" );
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document.write( "Ans:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Volume+=+36%2Asin%28theta%2F2%29%2Acos%28theta%2F2%29+ ALIGN=MIDDLE  ALT=\"+Volume+=+36%2Asin%28theta%2F2%29%2Acos%28theta%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ft%5E3+ ALIGN=MIDDLE  ALT=\"+ft%5E3+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "—<BR>\n" );
document.write( "To get the volume as a function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+theta+ ALIGN=MIDDLE  ALT=\"+theta+\"   DISABLED_style= \"max-width:90%; height:auto;\" > apply the half-angle formula:\r<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sin%28alpha%29+=+2%2Asin%28alpha%2F2%29%2Acos%28alpha%2F2%29+ ALIGN=MIDDLE  ALT=\"+sin%28alpha%29+=+2%2Asin%28alpha%2F2%29%2Acos%28alpha%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "—\r<BR>\n" );
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document.write( "Ans:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Volume+=+18%2Asin%28theta%29+ ALIGN=MIDDLE  ALT=\"+Volume+=+18%2Asin%28theta%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ft%5E3+ ALIGN=MIDDLE  ALT=\"+ft%5E3+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ft%5E3+ ALIGN=MIDDLE  ALT=\"+ft%5E3+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "The maximum volume will be at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+theta=90+degrees ALIGN=MIDDLE  ALT=\"+theta=90+degrees\"   DISABLED_style= \"max-width:90%; height:auto;\" >  where sin( ) has its max value of 1, which gives a volume of 18 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ft%5E3 ALIGN=MIDDLE  ALT=\"+ft%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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