Question

Determine the volume of the rectangular prism if it were dilated using a scale factor of 4.
length: 4
width: 3
height: 2
A. 1,536 m^3
B. 96 m^3
C. 6 m^3
D. 384 m^3

Answer 1

To determine the volume of the rectangular prism after a dilation, we start by calculating the volume of the original rectangular prism using the formula for the volume of a rectangular prism:

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

Given:
- Length = 4 m
- Width = 3 m
- Height = 2 m

Calculating the volume:

\[
\text{Volume} = 4 \times 3 \times 2 = 24 \text{ m}^3
\]

Next, we apply the dilation with a scale factor of 4. When a three-dimensional shape is dilated by a scale factor \( k \), the volume of the new shape is given by the formula:

\[
\text{New Volume} = k^3 \times \text{Original Volume}
\]

In this case, \( k = 4 \) and the original volume is \( 24 \text{ m}^3 \):

\[
\text{New Volume} = 4^3 \times 24 = 64 \times 24
\]

Calculating \( 64 \times 24 \):

\[
64 \times 24 = 1536 \text{ m}^3
\]

Thus, the volume of the rectangular prism after dilation is:

\[
\boxed{1536 \text{ m}^3}
\]

So the correct answer is A. 1,536 m³.