document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175766>Question 1175766</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 cu. m. Find the lateral surface area of the pyramid. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #801517 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=801517\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let s be the side length of the equilateral triangle which is the base of the pyramid.<br><BR>\n" );
document.write( "The given slant height of 9 is the hypotenuse of a right triangle in which the legs are the height h of the pyramid and a segment from the center of the triangular base to the middle of one edge.  The segment from the center of the base to the middle of one edge is 1/3 of the altitude of the triangular base, which is sqrt(3)/2 times the side length s.  That makes the length of that segment <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%2Asqrt%283%29%2F6 ALIGN=MIDDLE  ALT=\"s%2Asqrt%283%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "The Pythagorean Theorem with that right triangle then gives us the equation<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%5E2+=+h%5E2%2B%28s%2Asqrt%283%29%2F6%29%5E2 ALIGN=MIDDLE  ALT=\"9%5E2+=+h%5E2%2B%28s%2Asqrt%283%29%2F6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81+=+h%5E2%2Bs%5E2%2F12 ALIGN=MIDDLE  ALT=\"81+=+h%5E2%2Bs%5E2%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [1]<br><BR>\n" );
document.write( "The volume of the pyramid is given as 50.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%281%2F3%29Bh ALIGN=MIDDLE  ALT=\"V+=+%281%2F3%29Bh\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "where B is the area of the base.  The base is an equilateral triangle with side length s; its area is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2%2Asqrt%283%29%2F4 ALIGN=MIDDLE  ALT=\"s%5E2%2Asqrt%283%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  So<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%281%2F3%29%28s%5E2%2Asqrt%283%29%2F4%29%28h%29 ALIGN=MIDDLE  ALT=\"V+=+%281%2F3%29%28s%5E2%2Asqrt%283%29%2F4%29%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So now we have the equation<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50+=+%281%2F3%29%28s%5E2%2Asqrt%283%29%2F4%29%28h%29 ALIGN=MIDDLE  ALT=\"50+=+%281%2F3%29%28s%5E2%2Asqrt%283%29%2F4%29%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [2]<br><BR>\n" );
document.write( "s is a parameter we are using in our analysis; our objective is to find the height, h.  To do that, we need to eliminate s (or, actually, s^2) between the two equations we have.<br><BR>\n" );
document.write( "Solve [2] for s^2... <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50+=+%281%2F3%29%28s%5E2%2Asqrt%283%29%2F4%29%28h%29 ALIGN=MIDDLE  ALT=\"50+=+%281%2F3%29%28s%5E2%2Asqrt%283%29%2F4%29%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2+=+600%2Fsqrt%283%29%28h%29+=+200%2Asqrt%283%29%28h%29 ALIGN=MIDDLE  ALT=\"s%5E2+=+600%2Fsqrt%283%29%28h%29+=+200%2Asqrt%283%29%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [3]<br><BR>\n" );
document.write( "... and substitute in [1]<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81+=+h%5E2%2B%2850%2F3%29%2Asqrt%283%29%28h%29 ALIGN=MIDDLE  ALT=\"81+=+h%5E2%2B%2850%2F3%29%2Asqrt%283%29%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [4]<br><BR>\n" );
document.write( "That's the ugly part of the analysis. I leave the rest to you.<br><BR>\n" );
document.write( "(1) Use a graphing calculator or some other utility to solve equation [4] for the height h.<BR>\n" );
document.write( "(2) Use that value of h in [3] to find the side length s.<BR>\n" );
document.write( "(3) The lateral surface area is the area of three congruent isosceles triangles each with height 9 and base length s.<br><BR>\n" );
document.write( "Note you can verify your solution by seeing that the values you find for h and s satisfy equation [1].<br><BR>\n" );
document.write( " </SPAN>\n" );
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