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The new volume is two times the square root of two, in cubic feet. That is: approximately 2.83 cubic feet.\
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Explanation
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To double the surface area of any regular polyhedron such as a cube, you must increase the lengths of its edges by a factor of the square root of two, because the surface area is the edge length squared. Hence the volume increases by a factor of "the square root of two, cubed," which is two times the square root of two.\
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Calculating the new edge length is not necessary because for any regular polyhedron (as well as for any three-dimensional object, so far as I am aware), an increase in surface area by any factor x is always accompanied by an increase in volume by a factor x^(1.5).}