Find the volume of a cube if the area of its diagonal section is 64√2 dm2.

Let the length of the edge of the cube be X cm.

All side faces and bases of the cube are squares.

In a right-angled triangle ACD, according to the Pythagorean theorem, we determine the length of the hypotenuse AС.

AC ^ 2 = AD ^ 2 +СD ^ 2 = X ^ 2 + X ^ 2 = 2 * X ^ 2.

AC = X * √2 cm.

The diagonal section АА1С1С is a rectangle, then Ssection = АА1 * АС = X * X * √2 = X2 * √2 cm2.

X ^ 2 * √2 = 64 * √2.

X ^ 2 = 64.

X = 8 cm.

Then the volume of the cube is: V = X ^ 3 = 83 = 512 cm3.

Answer: The volume of the cube is 512 cm3.