document.write( "Question 1027670: If the surface of a cube is 6x^2 -36x +54, what is the expression for the volume of the cube? \n" ); document.write( "

Algebra.Com's Answer #853890 by ikleyn(53742) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "If the surface of a cube is 6x^2 -36x +54, what is the expression for the volume of the cube?
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\n" ); document.write( "\n" ); document.write( " I will give different solution from that by @mananth and from that by @Theo,\r
\n" ); document.write( "\n" ); document.write( " and at the end I will explain why the solutions by both these tutors are incorrect and why they should be changed.\r
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document.write( "The surface area of the cube is given by the polynomial expression  6x^2 - 36x + 54.\r\n" );
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document.write( "It is the combined area of 6 identical faces of the cube.  \r\n" );
document.write( "The area of each face separately is\r\n" );
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document.write( "    (6x^2 - 36x + 54)/6 = x^2 - 12x + 9 = (x-3)^2.\r\n" );
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document.write( "Notice that the given polynomial is always non-negative, \r\n" );
document.write( "so its values make sense as the face area for all values of x =/= 3.\r\n" );
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document.write( "The length of the side of this cube is   = |x-3|.\r\n" );
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document.write( "Notice that I use the absolute value  |x-3|  here, \r\n" );
document.write( "as the length is always assumed to be non-negative.\r\n" );
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document.write( "The volume of the cube is then  .\r\n" );
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document.write( "This formula works for all values  x =/= 3, including  x < 3, \r\n" );
document.write( "while the formulas in the posts by @mananth and by @Theo do not work at x < 3 (!)\r\n" );
document.write( "giving negative values for the edge length and for the volume.\r\n" );
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document.write( "So, my solution is MORE UNIVERSAL and more ACCURATE than the @mananth' solution and/or than the @Theo' solution.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Actually, this seemingly simple task has a $\"highlight%28highlight%28well%29%29\"$ $\"highlight%28highlight%28hidden%29%29\"$ murderous $\"highlight%28highlight%28trap%29%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "My solution discloses the trap, teaches you on how to avoid this trap
\n" ); document.write( "and also teaches you to be always aware and always accurate in Math.\r
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It is I N T E R E S T I N G

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\n" ); document.write( "\n" ); document.write( "Today, on Feb.14,2026, I submitted this problem to GOOGLE AI Overview to see how it treats it.\r
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\n" ); document.write( "\n" ); document.write( "Google AI Overview practically repeated incorrect/incomplete solution by @mananth and by @Theo.\r
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\n" ); document.write( "\n" ); document.write( "Naturally, I reported to Google AI about their fault through their feedback system.\r
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\n" ); document.write( "\n" ); document.write( "Then I made my next experiment. I submitted the same problem to other AI,
\n" ); document.write( "math-gpt.org , which (i think) is slightly more advanced.\r
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\n" ); document.write( "\n" ); document.write( "This other AI repeated the same tediousness and produced the same \"lame\" solution
\n" ); document.write( "(which reminds me a lame horse with three legs).\r
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\n" ); document.write( "\n" ); document.write( "It is the real world of AI in the area of solving school Math problems, in which we all live now.\r
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