The diagonal of a cube is equal to the diagonal of the face of the other cube. Find the ratio of their volumes.

Let the diagonal length of the first cube be X cm.
Let’s construct the BC diagonal at the base of the cube.
The ABC triangle is rectangular and isosceles, then BC ^ 2 = 2 * AB ^ 2.
In a right-angled triangle CBB1, BB1 = AB, then, by the Pythagorean theorem:
B1C ^ 2 = AB ^ 2 + 2 * AB ^ 2.
X ^ 2 = 3 * AB ^ 2.
AB = X / √3 cm.
By condition, the side of the second where is equal to the diagonal of the first cube. KM = X cm.
Then V ^ 2 = X ^ 3 cm3.
V1 = AB ^ 3 = X3 / 3 * √3 cm3.
Then V2 / V1 = X ^ 3 / (X ^ 3/3 * √3) = 3 * √3.
Answer: The ratio of volumes of cubes is 3 * √3.