The volume of the cube is 12 √12, find its diagonal.

Let the cube ABCDA1B1C1D1 be given, where ABCD is the lower base, and A1B1C1D1 is the upper base. Let the side of the cube be k. The volume of the cube is equal to k ^ 3 = k * k * k.

1) Calculate the side of the cube k:

k ^ 3 = 12 * √12,, k = (k ^ 3) ^ (1/3) = (12 * √12) ^ 1/3 = (12 * 12 ^ 1/2) ^ 1/3 = [ 12 ^ (3/2)] ^ 1/3 = 12 ^ (3/2) * (1/3) = 12 ^ 1/2 = √12.

2) The diagonal of the cube – d – is a segment from some point of the lower base to the most distant point of the upper base, that is, it is a segment AC1.

Find AC1 from triangle ACC1: AC1 = q = √ (AC ^ 2 + CC1 ^ 2) =

√ (AB ^ 2 + BC ^ 2 + CC1 ^ 2) = √ (3 * K ^ 2) = √ [3 * √12) ^ 2] = √36 = 6.