Question

if the rectangular prism 4m 18m 5m is dilated using a scale factor of 1/2, what would be the volume of the dilated rectangular prism?

Answer 1

To find the volume of a rectangular prism, you multiply the length, width, and height together.

Given the dimensions of the original rectangular prism are:
- Length = 4 m
- Width = 18 m
- Height = 5 m

The volume \( V \) of the original rectangular prism is calculated as follows:

\[
V = \text{Length} \times \text{Width} \times \text{Height} = 4 \, \text{m} \times 18 \, \text{m} \times 5 \, \text{m}
\]

Calculating this gives:

\[
V = 4 \times 18 = 72
\]
\[
V = 72 \times 5 = 360 \, \text{m}^3
\]

Now, when the prism is dilated using a scale factor of \( \frac{1}{2} \), each dimension of the prism is multiplied by \( \frac{1}{2} \):

- New Length = \( 4 \times \frac{1}{2} = 2 \, \text{m} \)
- New Width = \( 18 \times \frac{1}{2} = 9 \, \text{m} \)
- New Height = \( 5 \times \frac{1}{2} = 2.5 \, \text{m} \)

The volume \( V' \) of the dilated rectangular prism is then:

\[
V' = \text{New Length} \times \text{New Width} \times \text{New Height} = 2 \, \text{m} \times 9 \, \text{m} \times 2.5 \, \text{m}
\]

Calculating this gives:

\[
V' = 2 \times 9 = 18
\]
\[
V' = 18 \times 2.5 = 45 \, \text{m}^3
\]

Thus, the volume of the dilated rectangular prism is \(\boxed{45} \, \text{m}^3\).