Question

determine the volume of the rectangular prism B4m L3m H2m if it were dilated using a scale factor of 4

Answer 1

To find the volume of the rectangular prism before dilation, use the formula for the volume of a rectangular prism:

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

Given the dimensions:
- Length (L) = 4 m
- Width (W) = 3 m
- Height (H) = 2 m

The original volume is calculated as follows:

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

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

- New Length = \(4 \times 4 = 16 \, \text{m}\)
- New Width = \(4 \times 3 = 12 \, \text{m}\)
- New Height = \(4 \times 2 = 8 \, \text{m}\)

Now, calculate the volume of the dilated rectangular prism:

\[
\text{New Volume} = 16 \, \text{m} \times 12 \, \text{m} \times 8 \, \text{m}
\]

Calculating this gives:

\[
\text{New Volume} = 16 \times 12 = 192 \, \text{m}^2
\]
\[
\text{New Volume} = 192 \times 8 = 1536 \, \text{m}^3
\]

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

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

Alternatively, you can find the new volume by using the property of dilations in three dimensions:

The volume scales by the cube of the scale factor. Therefore, with a scale factor of 4:

\[
\text{New Volume} = \text{Original Volume} \times \text{Scale Factor}^3
\]
\[
\text{New Volume} = 24 \, \text{m}^3 \times 4^3 = 24 \, \text{m}^3 \times 64 = 1536 \, \text{m}^3
\]

Both methods confirm that the new volume is:

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