Question

Determine the volume of the rectangular prism if it were dilated using a scale factor of 4. (1 point) Responses 6 m3 6 m cubed 96 m3 96 m cubed 384 m3 384 m cubed 1,536 m3
height 2
width 8
length 4

Answer 1

To determine the volume of the rectangular prism and then find the volume after dilating it with a scale factor of 4, we first calculate the original volume.

The formula for the volume \( V \) of a rectangular prism is:
\[
V = \text{length} \times \text{width} \times \text{height}
\]

Given:
- Height = 2 m
- Width = 8 m
- Length = 4 m

Let's calculate the original volume:

\[
V = 4 \, \text{m} \times 8 \, \text{m} \times 2 \, \text{m} = 64 \, \text{m}^3
\]

Next, when a shape is dilated by a scale factor \( k \), the new volume \( V' \) is given by:
\[
V' = k^3 \times V
\]
In this case, the scale factor \( k = 4 \).

So we calculate:
\[
V' = 4^3 \times 64 \, \text{m}^3
\]
Calculating \( 4^3 \):
\[
4^3 = 64
\]
Now, substitute this back into the volume formula:
\[
V' = 64 \times 64 = 4096 \, \text{m}^3
\]

However, my calculations above are incorrect. The volume after dilation should be:

- Original Volume \( = 64 \, \text{m}^3 \)
- Scale Factor \( = 4 \), so \( 4^3 = 64 \)

Hence:
\[
V' = 64 \times 64 = 4096 \text{ m}^3
\]

Therefore, **the volume of the rectangular prism after dilation is 4096 m³.**