Let's start by finding the length of one edge of the original cube.
The surface area of a cube can be expressed as 6s^2, where s is the length of one edge.
So we have:
6s^2 = 2400
Solving for s, we get:
s = sqrt(2400/6)
s = 20
Therefore, the original cube has an edge length of 20 cm.
To find the volume of the similar box enlarged by a scale factor of 1.5, we need to multiply the volume of the original cube by (1.5)^3, since the scale factor applies to all three dimensions (length, width, and height).
The volume of the original cube is:
V = s^3 = 20^3 = 8000 cm³
So the volume of the similar box is:
V' = V x (1.5)^3 = 8000 x 1.5^3 = 8000 x 3.375 = 27000 cm³
Therefore, the volume of the similar box enlarged by a scale factor of 1.5 is 27,000 cm³.