Question

A rectangular prism has dimensions 2 feet by 4 feet by 8 feet. A cube has the same volume. What is the length of each of the cube's sides?

Answer 1

To find the length of each side of the cube that has the same volume as the rectangular prism, we first need to calculate the volume of the rectangular prism.

The volume \( V \) of a rectangular prism is given by the formula:

\[
V = \text{length} \times \text{width} \times \text{height}
\]

Substituting the given dimensions of the prism:

\[
V = 2 \, \text{feet} \times 4 \, \text{feet} \times 8 \, \text{feet}
\]

Calculating that:

\[
V = 2 \times 4 = 8 \quad \text{(first two dimensions)}
\]
\[
V = 8 \times 8 = 64 \quad \text{(including third dimension)}
\]

So, the volume of the rectangular prism is \( 64 \, \text{cubic feet} \).

Next, we need to find the side length of a cube that has the same volume. The volume \( V \) of a cube is given by the formula:

\[
V = s^3
\]

where \( s \) is the length of a side of the cube. We know the volume \( V \) is \( 64 \, \text{cubic feet} \), hence:

\[
s^3 = 64
\]

To find \( s \), we take the cube root of both sides:

\[
s = \sqrt[3]{64}
\]

Calculating \( \sqrt[3]{64} \):

\[
64 = 4^3 \quad \text{(since } 4 \times 4 \times 4 = 64\text{)}
\]
Thus:

\[
s = 4 \, \text{feet}
\]

Therefore, the length of each side of the cube is

\[
\boxed{4} \, \text{feet}
\]