document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1137885>Question 1137885</A>: <I><SPAN STYLE=\"word-break: break-all;\"> All of the faces of a rectangular block are painted red. The rectangular block is cut into unit cubes with a volume of 1 cm3 each. It is known that there are 30 unit cubes with none of their faces painted red and x unit cubes each with exactly two faces painted red.\r<BR>\n" );
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document.write( "Find the sum of all possible values of x. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #755757 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=755757\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let the dimensions of the rectangular block be (a+2), (b+2), and (c+2).  Then the unit cubes with no faces painted form a block with dimensions a, b, and c.<br><BR>\n" );
document.write( "The number of cubes with no faces painted is to be 30, so the product abc is equal to 30.  There are 5 sets of integer dimensions with that product:<br><BR>\n" );
document.write( "1x1x30<BR>\n" );
document.write( "1x2x15<BR>\n" );
document.write( "1x3x10<BR>\n" );
document.write( "1x5x6<BR>\n" );
document.write( "2x3x5<br><BR>\n" );
document.write( "On the rectangular block, the unit cubes with 2 faces painted are on the edges, but not on the corners.  There are 12 edges on a rectangular solid.  In our rectangular solid with dimensions (a+2), (b+2), and (c+2), there are 4 edges with length a (not counting the corners), 4 edges with length b (not counting the corners), and 4 edges with length c (not counting the corners).<br><BR>\n" );
document.write( "So, given the dimensions a, b, and c of the block of unit cubes with no faces painted, the number of units cubes with 2 faces painted is 4(a+b+c).<br><BR>\n" );
document.write( "With the analysis of the problem complete, we now just plug in numbers to find the answer to the problem.<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "(1) abc = 1*1*30 --> 4(a+b+c) = 4(32) = 128\r\n" );
document.write( "(2) abc = 1*2*15 --> 4(a+b+c) = 4(18) =  72\r\n" );
document.write( "(3) abc = 1*3*10 --> 4(a+b+c) = 4(14) =  56\r\n" );
document.write( "(4) abc = 1*5*6  --> 4(a+b+c) = 4(12) =  48\r\n" );
document.write( "(5) abc = 2*3*5  --> 4(a+b+c) = 4(10) =  40\r\n" );
document.write( "                                       -----\r\n" );
document.write( "                                        344<br></PRE><BR>\n" );
document.write( "ANSWER: 344 </SPAN>\n" );
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