Question
Determine the volume of the rectangular prism if it were dilated using a scale factor of 4. the rectangular prism has a height of 2 meters, a width of 3 meters, and a length of 4 meters. (1 point) Responses 1,536 m3 1,536 m cubed 96 m3 96 m cubed 6 m3 6 m cubed 384 m3
Answers
Answered by
GPT-4o mini
To find the volume of the rectangular prism after dilation with a scale factor of 4, we first need to calculate the original volume of the prism.
The formula for the volume \( V \) of a rectangular prism is:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
Given:
- Height = 2 meters
- Width = 3 meters
- Length = 4 meters
Calculating the original volume:
\[
V = 4 \, \text{m} \times 3 \, \text{m} \times 2 \, \text{m} = 24 \, \text{m}^3
\]
Now we apply the dilation. When a shape is dilated by a scale factor of \( k \), the new volume \( V' \) can be calculated by multiplying the original volume by \( k^3 \):
\[
V' = k^3 \times V
\]
In this case, the scale factor \( k = 4 \):
\[
V' = 4^3 \times 24 \, \text{m}^3 = 64 \times 24 \, \text{m}^3
\]
Calculating \( 64 \times 24 \):
\[
64 \times 24 = 1536 \, \text{m}^3
\]
Thus, the volume of the rectangular prism after dilation is **1,536 m³**.
The formula for the volume \( V \) of a rectangular prism is:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
Given:
- Height = 2 meters
- Width = 3 meters
- Length = 4 meters
Calculating the original volume:
\[
V = 4 \, \text{m} \times 3 \, \text{m} \times 2 \, \text{m} = 24 \, \text{m}^3
\]
Now we apply the dilation. When a shape is dilated by a scale factor of \( k \), the new volume \( V' \) can be calculated by multiplying the original volume by \( k^3 \):
\[
V' = k^3 \times V
\]
In this case, the scale factor \( k = 4 \):
\[
V' = 4^3 \times 24 \, \text{m}^3 = 64 \times 24 \, \text{m}^3
\]
Calculating \( 64 \times 24 \):
\[
64 \times 24 = 1536 \, \text{m}^3
\]
Thus, the volume of the rectangular prism after dilation is **1,536 m³**.