Nested Cuboids with 1/8 Ratio: Calculating Volume Relationship

To solve this problem, we need to determine how many smaller cuboids fit into the larger cuboid given that each side of the smaller cuboid is one-eighth the length of the corresponding side of the larger cuboid.

First, let's calculate the scaling effect on volume:

• The length $AB$ of the smaller cuboid is $\frac{1}{8}$ of length $AC$ of the larger cuboid.
• The width $BF$ of the smaller cuboid is $\frac{1}{8}$ of width $CG$ of the larger cuboid.
• The height $AD$ of the smaller cuboid is $\frac{1}{8}$ of height $AE$ of the larger cuboid.

The volume of a cuboid is given by multiplying its three dimensions (length, width, and height). Thus, the volume of the smaller cuboid is:

$\left(\frac{1}{8} \times AC\right) \times \left(\frac{1}{8} \times CG\right) \times \left(\frac{1}{8} \times AE\right)$

$= \frac{1}{8} \times \frac{1}{8} \times \frac{1}{8} \times (AC \times CG \times AE)$

$= \frac{1}{8^3} \times (\text{volume of the larger cuboid})$

$= \frac{1}{512} \times (\text{volume of the larger cuboid})$

Therefore, the volume of the smaller cuboid is $\frac{1}{512}$ of the larger cuboid's volume. This indicates that:

$512$ smaller cuboids fit into the larger cuboid.

Therefore, the number of times the small cuboid fits inside the larger cuboid is 512.