document.write( "Question 1099031: The surface area and diagonal of a cuboid are 288 sq.cm and 12 cm respectively. Show that the cuboid is a cube. \n" ); document.write( "

Algebra.Com's Answer #713474 by ikleyn(52787) $\"\"$ $\"About$

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\n" ); document.write( "This condition means EXACTLY THE FOLLOWING:\r
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document.write( "    We are given a rectangular box (a PRISM) with dimensions \"a\", \"b\" and \"c\".\r\n" );
document.write( "    The surface area of the box is 288 sq. cm.\r\n" );
document.write( "    The 3D diagonal of the box is 12 cm long.\r\n" );
document.write( "    Prove that the box is a cube.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Therefore, the solution by the other tutor is IRRELEVANT.\r
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\n" ); document.write( "\n" ); document.write( "The correct solution is THIS:\r
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document.write( "    Let us consider the quadratic form \r\n" );
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document.write( "        Q(a,b,c) = .     (1)\r\n" );
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document.write( "     We have Q(a,b,c) =  +  +  = \r\n" );
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document.write( "                      =  - .   (2)\r\n" );
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document.write( "     As everybody knows  (or MUST know),\r\n" );
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document.write( "         is the square of the length of the 3D diagonal, so it is equal to  cm^2.\r\n" );
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document.write( "     Therefore the term    in (2) is equal to   = 288 cm^2.\r\n" );
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document.write( "     Again, as everybody knows (or MUST know),\r\n" );
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document.write( "          is the surface area of the rectangular prism.\r\n" );
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document.write( "     Therefore,  the term    in  (2)  has the given value of 288 cm^2 under the given condition.\r\n" );
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document.write( "     Now it is clear that at given data the form  Q(a,b,c)  is equal to zero.\r\n" );
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document.write( "     But  Q(a,b,c) is the sum of squares, and can be equal to zero if and only if a = b,  b = c  and a = c.\r\n" );
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document.write( "     It implies that  a = b = c  and the prism is a cube under given condition.\r\n" );
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