document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1155822>Question 1155822</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Cube A is inscribed in sphere B, which is inscribed in cube C. If the sides of cube A have length 4, what is the volume of cube C? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #778497 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=778497\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Edit: I realized I made an error, but I have fixed the solution below (this is more or less a complete rewrite compared to my original solution).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Diagram of cube A:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/dp1xJ2u.png\"><BR>\n" );
document.write( "The left shows a 2D flat view of the bottom face of the cube. This is square ABCD. The right shows the 3D version of cube A. At the center of cube A is point K\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I've added point M directly below point K such that M is on the bottom face of the cube. Point L bisects segment AB. Triangle ALM has legs of 2 each, so the hypotenuse is 2*sqrt(2) through the pythagorean theorem. So AM = 2*sqrt(2)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Triangle AMK has legs AM = 2*sqrt(2) and MK = 2. Using the pythagorean theorem again has us get AK = 2*sqrt(3). Therefore, segment AH = 2*(2*sqrt(3)) = 4*sqrt(3). You can use the space diagonal formula or the distance formula to find the length of AH. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The radius of sphere B is 4*sqrt(3) units long.<BR>\n" );
document.write( "The side length of cube C is 4*sqrt(3) units long. This is so sphere B fits snugly inside cube C. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Volume of cube = (side length)^3<BR>\n" );
document.write( "Volume of cube C = (4*sqrt(3))^3<BR>\n" );
document.write( "Volume of cube C = (4*sqrt(3))(4*sqrt(3))(4*sqrt(3))<BR>\n" );
document.write( "Volume of cube C = [(4*sqrt(3))(4*sqrt(3))](4*sqrt(3))<BR>\n" );
document.write( "Volume of cube C = (4*4*sqrt(3)*sqrt(3))(4*sqrt(3))<BR>\n" );
document.write( "Volume of cube C = (16*3)(4*sqrt(3))<BR>\n" );
document.write( "Volume of cube C = 48(4*sqrt(3))<BR>\n" );
document.write( "Volume of cube C = 192*sqrt(3)<BR>\n" );
document.write( "Volume of cube C = 332.553755053224<BR>\n" );
document.write( "Use your calculator to compute the approximate value in the last step.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Exact Answer: <font color=red>192*sqrt(3) cubic units</font><BR>\n" );
document.write( "Approximate Answer: <font color=red>332.553755053224 cubic units</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );