document.write( "Question 469438: a rectangular solid of maximum volume is to be cut from a solid sphere of radius r.determine the dimensions L , H and the volume of the solid formed.
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Algebra.Com's Answer #322077 by ccs2011(207) $\"\"$ $\"About$

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Maximum volume given same 3-d space is achieved by a cube of equal side lengths.
\n" ); document.write( "Same logic as a square maximizes area given same perimeter.
\n" ); document.write( "Goal is to find largest cube confined to inside of sphere of radius r.
\n" ); document.write( "To achieve this we want the corners of the cube to be touching the outer surface of the sphere.
\n" ); document.write( "In other words, the diagonal from one corner to the other should equal the diameter of the sphere.
\n" ); document.write( "Let x be the length of a side of the cube.
\n" ); document.write( "To find length from one corner to the other, first need to find diagonal of the base (square).
\n" ); document.write( "Diagonal of a square of length x is $\"sqrt%282%29\"$ x
\n" ); document.write( "Corner length is $\"sqrt%28x%5E2+%2B+%28sqrt%282%29x%29%5E2%29+=+sqrt%283x%5E2%29+=+sqrt%283%29x\"$
\n" ); document.write( "Now set this equal to diameter of sphere
\n" ); document.write( " $\"sqrt%283%29x+=+2r\"$
\n" ); document.write( " $\"x+=+2r%2Fsqrt%283%29+=+1.155r\"$
\n" ); document.write( "Volume is x^3
\n" ); document.write( " $\"V+=+%281.155r%29%5E3+=+1.54r%5E3\"$ \n" ); document.write( "

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