document.write( "Question 1198260: The altitude of a pyramid is 12 cm. A parallel to the base cut the pyramid into two solids of equal volumes. The distance of the cutting plane from the vertex is expressed as d = α^3√β cm where α and β are integers. Find the smallest sum of α and β. \n" ); document.write( "

Algebra.Com's Answer #831832 by ikleyn(52834) $\"\"$ $\"About$

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\n" ); document.write( "The altitude of a pyramid is 12 cm. A parallel to the base cut the pyramid into two solids of equal volumes.
\n" ); document.write( "The distance of the cutting plane from the vertex is expressed as d = α^3√β cm where α and β are integers.
\n" ); document.write( "Find the smallest sum of α and β.
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\n" ); document.write( "\n" ); document.write( "                Your formulation in the post is INCORRECT.
\n" ); document.write( "                I was so kind that re-wrote the problem in a right form.
\n" ); document.write( "                The correct version is as follows:\r
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document.write( "    The altitude of a pyramid is 12 cm. A parallel to the base cut the pyramid into two solids of equal volumes. \r\n" );
document.write( "    The distance of the cutting plane from the vertex is expressed as d =  cm, where α and β are integers. \r\n" );
document.write( "    Find the smallest sum of α and β.\r\n" );
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\n" ); document.write( "\n" ); document.write( " See my solution below for this corrected version.\r
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document.write( "Let x be the distance from the vertex to the cutting plane.\r\n" );
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document.write( "The cut part and the whole pyramid are similar solids, so the ratio\r\n" );
document.write( "of their volumes (which is 1:2) is the cube of the ratio of their \r\n" );
document.write( "respective linear elements.\r\n" );
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document.write( "It gives us this proportion\r\n" );
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document.write( "     = ,\r\n" );
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document.write( "or\r\n" );
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document.write( "    x^3 =  =  =  = .\r\n" );
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document.write( "Hence,  x =  = .\r\n" );
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document.write( "Thus  = 6,   = 4, and the sum   +  = 6 + 4 = 10.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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