In a rectangular parallelepiped, the edges are 2: 3: 6 and its diagonal is 42. Find the volume of the parallelepiped.

Let the lengths of the sides of the parallelepiped be 2 * X, 3 * X cm, 6 * X cm.

The square of the length of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of the lengths of its sides.

AC1 ^ 2 = (2 * X) ^ 2 + (3 * X) ^ 2 + (6 * X) ^ 2.

1764 = 4 * X ^ 2 + 9 * X ^ 2 + 36 * X ^ 2.

49 * X ^ 2 = 1764.

X ^ 2 = 1764/49 = 36.

X = 6.

Then the lengths of the sides of the parallelepiped are:

2 * 6 = 12 cm.

3 * 6 = 18 cm.

6 * 6 = 36 cm.

Determine the volume of the parallelepiped:

V = 12 * 18 * 36 = 7776 cm3.

Answer: The volume is 7776 cm3.