document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1117233>Question 1117233</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A regular triangular pyramid with a slant height of 9 m has a volume equal to 50 m^3. Find the lateral area of the pyramid.\r<BR>\n" );
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document.write( "Answer:84.65 sq.m.\r<BR>\n" );
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document.write( "How to solve? Please explain. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #732262 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=732262\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let the altitude of the pyramid be h.  Then consider the right triangle with that altitude as one leg and the slant height of 9 as the hypotenuse.  Then the other leg of that right triangle has length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2881-h%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%2881-h%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "That other leg of the right triangle is one-third the length of the altitude of the triangular base; the length of that altitude is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > times the side length s of the triangular base.  So<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%28sqrt%283%29%2F2%29+=+3%28sqrt%2881-h%5E2%29%29 ALIGN=MIDDLE  ALT=\"s%28sqrt%283%29%2F2%29+=+3%28sqrt%2881-h%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+%282%2Asqrt%283%29%29%28sqrt%2881-h%5E2%29%29 ALIGN=MIDDLE  ALT=\"s+=+%282%2Asqrt%283%29%29%28sqrt%2881-h%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We can find the height of the pyramid by knowing that its volume is 50:<br><BR>\n" );
document.write( "volume = one-third base times height<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50+=+%281%2F3%29%28%28s%5E2%29%28sqrt%283%29%2F4%29%29%28h%29 ALIGN=MIDDLE  ALT=\"50+=+%281%2F3%29%28%28s%5E2%29%28sqrt%283%29%2F4%29%29%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50+=+%281%2F3%29%28%2812%2881-h%5E2%29%29%28sqrt%283%29%2F4%29%29%28h%29 ALIGN=MIDDLE  ALT=\"50+=+%281%2F3%29%28%2812%2881-h%5E2%29%29%28sqrt%283%29%2F4%29%29%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50+=+%2881-h%5E2%29%28h%29%28sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"50+=+%2881-h%5E2%29%28h%29%28sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graphing calculator shows two solutions to that equation; to several decimal places the solutions are h=0.35695 and h=8.81621.<br><BR>\n" );
document.write( "So there will be two different pyramids that give the correct volume.  The one with the very small height is probably not the intended one, so from here I will assume the height is 8.81621.<br><BR>\n" );
document.write( "Substituting that value in the expression for the side length of the triangular base,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+%282%2Asqrt%283%29%29%28sqrt%2881-8.81621%5E2%29%29+=+6.26844 ALIGN=MIDDLE  ALT=\"s+=+%282%2Asqrt%283%29%29%28sqrt%2881-8.81621%5E2%29%29+=+6.26844\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now the lateral surface area consists of three congruent triangles, each with base 6.26822 and height 8.81621.  The total surface area is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%29%281%2F2%29%286.26822%29%288.81621%29+=+84.62389 ALIGN=MIDDLE  ALT=\"%283%29%281%2F2%29%286.26822%29%288.81621%29+=+84.62389\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This is very close to the given answer.  Since I kept 5 or more decimal places in all my calculations, I suspect that the given answer 84.65 was obtained keeping fewer decimal places in the calculations. </SPAN>\n" );
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