Find the volume of a rectangular parallelepiped if the sides of the base are 2 and 3, and the diagonal of the parallelepiped is √38

All faces of a rectangular parallelepiped are rectangles. It is known that the square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions, which means:

a ^ 2 + b ^ 2 + h ^ 2 = D ^ 2.

Hence:

h ^ 2 = D ^ 2 – a ^ 2 – b ^ 2 = (√38) ^ 2 – 2 ^ 2 – 3 ^ 2 = 38 – 4 – 9 = 25 = 52;

h = 5 – the height of the parallelepiped.

The volume of a parallelepiped is equal to the product of the area of the base by the height, and since the area of the base is equal to the product of the sides of the base, the volume of a rectangular parallelepiped is equal to the product of its three dimensions:

V = a * b * h = 2 * 3 * 5 = 30.