SOLUTION: 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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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.
Answer by ccs2011(207) About Me  (Show Source):
You can put this solution on YOUR website!
Maximum volume given same 3-d space is achieved by a cube of equal side lengths.
Same logic as a square maximizes area given same perimeter.
Goal is to find largest cube confined to inside of sphere of radius r.
To achieve this we want the corners of the cube to be touching the outer surface of the sphere.
In other words, the diagonal from one corner to the other should equal the diameter of the sphere.
Let x be the length of a side of the cube.
To find length from one corner to the other, first need to find diagonal of the base (square).
Diagonal of a square of length x is sqrt%282%29x
Corner length is sqrt%28x%5E2+%2B+%28sqrt%282%29x%29%5E2%29+=+sqrt%283x%5E2%29+=+sqrt%283%29x
Now set this equal to diameter of sphere
sqrt%283%29x+=+2r
x+=+2r%2Fsqrt%283%29+=+1.155r
Volume is x^3
V+=+%281.155r%29%5E3+=+1.54r%5E3