If a cube has an edge equal to the diagonal of another cube, find the ratio of their volumes.

SOURCE: One cube has a face equivalent to the total area

Let the sides of the cubes be a and b,

the length of the diagonal of the cube whose side is b is given as square root(3)*b

according to the given question,

a=square root(3)*b

the volume of the cube with side a is given as

v1=a^3=(square root(3)*b)^3

and the volume of the cube with side b is

v2=b^3

then their ratio of volumes is given as

v1/v2=3*square root(3)*b^3/b^3=3*square root(3)

i.e v1:v2=3*square root(3):1

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