Question

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

Answer 1

To determine the volume of the rectangular prism after a dilation, we first calculate the volume of the original prism using the formula for the volume of a rectangular prism, which is:

\[
\text{Volume} = \text{Height} \times \text{Width} \times \text{Length}
\]

Given the dimensions:
- Height = 2
- Width = 8
- Length = 4

Calculating the original volume:

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

Next, when the prism is dilated using a scale factor of 4, the dimensions of the prism will be multiplied by 4:

- New Height = \(2 \times 4 = 8\)
- New Width = \(8 \times 4 = 32\)
- New Length = \(4 \times 4 = 16\)

Now, we can find the new volume using the same formula:

\[
\text{New Volume} = \text{New Height} \times \text{New Width} \times \text{New Length}
\]

\[
\text{New Volume} = 8 \times 32 \times 16
\]

Calculating this step-by-step:

1. Calculate \(8 \times 32 = 256\)
2. Now multiply \(256 \times 16 = 4096\)

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

\[
\text{New Volume} = 4096 \text{ m}^3
\]

However, the scale factor affects the volume by the cube of the scale factor. We can also find the new volume using the cube of the scale factor:

\[
\text{Dilation Volume Factor} = \text{Scale Factor}^3 = 4^3 = 64
\]

The new volume can also be calculated as:

\[
\text{New Volume} = \text{Original Volume} \times 64 = 64 \text{ m}^3 \times 64 = 4096 \text{ m}^3
\]

So the final answer is:

**4096 m³**, which is not in the options provided. Ensure all options are correctly stated or a different calculation method was anticipated.