Question

A rectangular prism has a length of 6 cm, a width of 3 cm, and a height of 412cm.

The prism is filled with cubes that have edge lengths of 12 cm.



How many cubes are needed to fill the rectangular prism?

Enter your answer in the box.

To fill the rectangular prism,
cubes are needed.

Answer 1

To find out how many cubes are needed to fill the rectangular prism, we first need to calculate the volume of both the rectangular prism and the cubes.

1. **Volume of the rectangular prism**:
The volume \( V \) of a rectangular prism is calculated using the formula:
\[
V = \text{length} \times \text{width} \times \text{height}
\]

Given:
- Length = 6 cm
- Width = 3 cm
- Height = 412 cm

Calculating the volume:
\[
V = 6 \, \text{cm} \times 3 \, \text{cm} \times 412 \, \text{cm} = 6 \times 3 \times 412 = 7392 \, \text{cm}^3
\]

2. **Volume of a single cube**:
The volume of a cube with edge length \( s \) is given by:
\[
V = s^3
\]

Given:
- Edge length of the cube = 12 cm

Calculating the volume:
\[
V = 12 \, \text{cm} \times 12 \, \text{cm} \times 12 \, \text{cm} = 12^3 = 1728 \, \text{cm}^3
\]

3. **Number of cubes needed**:
To find out how many cubes are needed to fill the rectangular prism, divide the volume of the prism by the volume of one cube:
\[
\text{Number of cubes} = \frac{\text{Volume of the prism}}{\text{Volume of one cube}} = \frac{7392 \, \text{cm}^3}{1728 \, \text{cm}^3}
\]
Performing the division:
\[
\text{Number of cubes} = \frac{7392}{1728} = 4.28
\]

Since we cannot have a fraction of a cube, we need to consider that the number must be an integer. This means that in actuality, you cannot completely fill the rectangular prism with full cubes of 12 cm without wasting some space. Therefore, you can only fit a maximum of **4 cubes**, leaving some unused volume.

Thus, to fill the rectangular prism, only **4 cubes** are needed.

So, the answer is:

**4** cubes are needed.