document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1111892>Question 1111892</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The frustum has regular hexagon bases. The upper base measures 16 ft. on a side and the lower base measures 29 ft. on a side. The altitude of the frustum is 16 ft.  Find the mass of the frustum, if its density is 99 lbs. per cu. ft.\r<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 #726999 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=726999\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The answer from the other tutor is not correct.  With a frustum (truncated pyramid), you can't just use the length of a side halfway between the two bases to find the volume of the frustum....<br><BR>\n" );
document.write( "Here are three ways to find the volume of the frustum and thus find its weight.<br><BR>\n" );
document.write( "(1) Consider the frustum as a truncated hexagonal pyramid; find the volume of the original pyramid and subtract off the volume of the pyramid that was cut off.<br><BR>\n" );
document.write( "Use similar triangles to find the height h of the original pyramid:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28h-16%29%2F16+=+h%2F29 ALIGN=MIDDLE  ALT=\"%28h-16%29%2F16+=+h%2F29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=29h-464+=+16h ALIGN=MIDDLE  ALT=\"29h-464+=+16h\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13h+=+464 ALIGN=MIDDLE  ALT=\"13h+=+464\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+464%2F13 ALIGN=MIDDLE  ALT=\"h+=+464%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The volume of the original pyramid is one-third the area of the base, times the height; the area of the regular hexagonal base is the area of 6 equilateral triangles with side length 29:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%281%2F3%29%286%2A29%5E2%2Asqrt%283%29%2F4%29%2A%28464%2F13%29+=+25995.6844 ALIGN=MIDDLE  ALT=\"V+=+%281%2F3%29%286%2A29%5E2%2Asqrt%283%29%2F4%29%2A%28464%2F13%29+=+25995.6844\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The volume of the pyramid that is cut off is one-third the area of the base, times the height:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%281%2F3%29%286%2A16%5E2%2Asqrt%283%29%2F4%29%2A%28464%2F13-16%29+=+4365.834 ALIGN=MIDDLE  ALT=\"V+=+%281%2F3%29%286%2A16%5E2%2Asqrt%283%29%2F4%29%2A%28464%2F13-16%29+=+4365.834\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The volume of the frustum is the difference between the volumes of the two pyramids:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+25995.6844-4365.834+-+21629.85 ALIGN=MIDDLE  ALT=\"V+=+25995.6844-4365.834+-+21629.85\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(2) In that first method, we found the volume of the pyramid that was cut off directly, using the formula for the volume of a pyramid.  We can also find the volume of that pyramid using the fact that the original pyramid and the pyramid that was cut off are similar figures.<br><BR>\n" );
document.write( "The scale factor between the two pyramids is 16:29; the ratio of the volumes of the two pyramids is (16:29)^3.<br><BR>\n" );
document.write( "So the volume of the frustum can be calculated as the volume of the original pyramid, multiplied by (1 - (16/29)^3):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+25995.6844%281-%2816%2F29%29%5E3%29+=+21629.85 ALIGN=MIDDLE  ALT=\"V+=+25995.6844%281-%2816%2F29%29%5E3%29+=+21629.85\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(3) There is a rather obscure formula for the volume of a frustum of a pyramid with a regular polygonal base; for a frustum with a regular hexagonal base, with base side lengths 29 and 16 and height 16, the formula is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%281%2F3%29%28%286%2A29%5E2%2Asqrt%283%29%29%2F4%2B%286%2A29%2A16%2Asqrt%283%29%29%2F4%2B%286%2A16%5E2%2Asqrt%283%29%29%2F4%29%2816%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Evaluating this expression of course gives the same answer of 21629.85 for the volume of the frustum.<br><BR>\n" );
document.write( "That answer is in cubic feet; then to find the weight you simply multiply that by the density in pounds per cubic foot:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21629.85%2A99+=+2141355 ALIGN=MIDDLE  ALT=\"21629.85%2A99+=+2141355\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "2,141,355 pounds -- a rather heavy chunk of something.<BR>\n" );
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