Question

5 tall
8 long
4 wide

If the rectangular prism is dilated using a scale factor of 12 , what would be the volume of the dilated rectangular prism? (1 point)

Answer 1

To find the volume of a rectangular prism, you use the formula:

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

Given the dimensions:

- Height (tall) = 5
- Length (long) = 8
- Width = 4

First, calculate the volume of the original rectangular prism:

\[
\text{Volume} = 8 \times 4 \times 5
\]

Calculating this step-by-step:

\[
8 \times 4 = 32
\]

\[
32 \times 5 = 160
\]

So, the volume of the original rectangular prism is 160 cubic units.

Now, if the prism is dilated using a scale factor of 12, all dimensions (height, length, and width) are multiplied by 12. The new dimensions would be:

- New Height = \( 5 \times 12 = 60 \)
- New Length = \( 8 \times 12 = 96 \)
- New Width = \( 4 \times 12 = 48 \)

Now, we calculate the volume of the dilated rectangular prism:

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

\[
\text{Volume} = 96 \times 48 \times 60
\]

To simplify the calculation, we first find \( 96 \times 48 \):

\[
96 \times 48 = 4608
\]

Now multiply this by 60:

\[
4608 \times 60 = 276480
\]

Thus, the volume of the dilated rectangular prism is:

\[
\text{Volume} = 276480 \text{ cubic units}
\]

Alternatively, you can also find the volume by recognizing that if a prism is dilated by a scale factor of \( k \), the new volume is \( k^3 \) times the original volume. Here, \( k = 12 \), so:

\[
\text{New Volume} = 12^3 \times \text{Original Volume} = 1728 \times 160
\]

Calculating this gives:

\[
1728 \times 160 = 276480
\]

Thus, the final answer for the volume of the dilated rectangular prism is:

\[
\text{Volume} = 276480 \text{ cubic units}
\]