document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=567895>Question 567895</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Compute the volume of an icosahedron whose edge measures 3 inches. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #366856 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=366856\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The volume of a icosahedron (a regular polyhedron with 20 identical equilateral triangular faces, 12 vertices, and 30 edges) is given by the formula:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%285%2F12%29%2A%283+%2B+sqrt%285%29%29%2As%5E3 ALIGN=MIDDLE  ALT=\"V+=+%285%2F12%29%2A%283+%2B+sqrt%285%29%29%2As%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "in which V is the volume in cubic units and s is the length of an edge in units matching the cubic units of the volume.<BR>\n" );
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document.write( "For this problem you are given that the length of a side is 3 inches. Therefore, the volume calculated by using the formula will in units of cubic inches.<BR>\n" );
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document.write( "Start with the formula for the volume:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%285%2F12%29%2A%283+%2B+sqrt%285%29%29%2As%5E3 ALIGN=MIDDLE  ALT=\"V+=+%285%2F12%29%2A%283+%2B+sqrt%285%29%29%2As%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substitute 3 for s to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%285%2F12%29%2A%283+%2B+sqrt%285%29%29%2A3%5E3 ALIGN=MIDDLE  ALT=\"V+=+%285%2F12%29%2A%283+%2B+sqrt%285%29%29%2A3%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Replace the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E3 ALIGN=MIDDLE  ALT=\"3%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > with its equivalent value of 27:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%285%2F12%29%2A%283+%2B+sqrt%285%29%29%2A27 ALIGN=MIDDLE  ALT=\"V+=+%285%2F12%29%2A%283+%2B+sqrt%285%29%29%2A27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that the 12 in the denominator of the leading term can be replaced by 3 times 4 to result in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%285%2F%283%2A4%29%29%2A%283+%2B+sqrt%285%29%29%2A27 ALIGN=MIDDLE  ALT=\"V+=+%285%2F%283%2A4%29%29%2A%283+%2B+sqrt%285%29%29%2A27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply the leading term by the last term (27) to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%28%2827%2A5%29%2F%283%2A4%29%29%2A%283+%2B+sqrt%285%29%29 ALIGN=MIDDLE  ALT=\"V+=+%28%2827%2A5%29%2F%283%2A4%29%29%2A%283+%2B+sqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Divide the 3 in the denominator of the first term into the 27 in the numerator. This reduces the first term as shown:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%28%289%2A5%29%2F4%29%2A%283+%2B+sqrt%285%29%29 ALIGN=MIDDLE  ALT=\"V+=+%28%289%2A5%29%2F4%29%2A%283+%2B+sqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and multiply out the numerator of the first term and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%2845%2F4%29%2A%283+%2B+sqrt%285%29%29 ALIGN=MIDDLE  ALT=\"V+=+%2845%2F4%29%2A%283+%2B+sqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and if you prefer, you can multiply the right side to have the form of the answer be:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%2845%2A%283+%2B+sqrt%285%29%29%29%2F4 ALIGN=MIDDLE  ALT=\"V+=+%2845%2A%283+%2B+sqrt%285%29%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then by doing the distributed multiplication in the numerator you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%2845%2A%283+%2B+sqrt%285%29%29%29%2F4 ALIGN=MIDDLE  ALT=\"V+=+%2845%2A%283+%2B+sqrt%285%29%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The square root of 5 is an irrational number. Therefore, there is not an exact answer to this problem. However, you can approximate the answer by letting the square root of 5 equal 2.23606798. Substitute that value for the square root of 5 and the formula becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%2845%2A%283+%2B+2.23606798%29%29%2F4 ALIGN=MIDDLE  ALT=\"V+=+%2845%2A%283+%2B+2.23606798%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Add the terms in the parentheses at it simplifies to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+%2845%2A5.23606798%29%2F4 ALIGN=MIDDLE  ALT=\"V+=+%2845%2A5.23606798%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiplying out the numerator (45*5.23606798) and then dividing by the denominator (4) gives the \"approximate\" answer:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V+=+58.9057647 ALIGN=MIDDLE  ALT=\"V+=+58.9057647\"   DISABLED_style= \"max-width:90%; height:auto;\" > and its units are cubic inches<BR>\n" );
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document.write( "I hope this gives you the information you need to solve this problem. <BR>\n" );
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