Question
2 m
3 m
4 m
Determine the volume of the rectangular prism if it were dilated using a scale factor of 4.
a. 6 m^3
b. 96 m^3
c. 384 m^3
d. 1,536 m^3
3 m
4 m
Determine the volume of the rectangular prism if it were dilated using a scale factor of 4.
a. 6 m^3
b. 96 m^3
c. 384 m^3
d. 1,536 m^3
Answers
GPT-4o mini
To determine the volume of the rectangular prism after a dilation with a scale factor of 4, we first need to find the original volume of the prism with dimensions 2 m, 3 m, and 4 m.
The volume \( V \) of a rectangular prism is given by the formula:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
Substituting the given dimensions:
\[
V = 2 \, \text{m} \times 3 \, \text{m} \times 4 \, \text{m} = 24 \, \text{m}^3
\]
When a three-dimensional shape like a rectangular prism is dilated by a scale factor \( k \), the volume is affected by a factor of \( k^3 \):
\[
\text{New Volume} = \text{Original Volume} \times k^3
\]
Here, the scale factor \( k = 4 \):
\[
k^3 = 4^3 = 64
\]
Now to find the new volume:
\[
\text{New Volume} = 24 \, \text{m}^3 \times 64 = 1536 \, \text{m}^3
\]
Therefore, the volume of the rectangular prism after dilation is:
\[
\boxed{1,536 \, \text{m}^3}
\]
This matches option d.
The volume \( V \) of a rectangular prism is given by the formula:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
Substituting the given dimensions:
\[
V = 2 \, \text{m} \times 3 \, \text{m} \times 4 \, \text{m} = 24 \, \text{m}^3
\]
When a three-dimensional shape like a rectangular prism is dilated by a scale factor \( k \), the volume is affected by a factor of \( k^3 \):
\[
\text{New Volume} = \text{Original Volume} \times k^3
\]
Here, the scale factor \( k = 4 \):
\[
k^3 = 4^3 = 64
\]
Now to find the new volume:
\[
\text{New Volume} = 24 \, \text{m}^3 \times 64 = 1536 \, \text{m}^3
\]
Therefore, the volume of the rectangular prism after dilation is:
\[
\boxed{1,536 \, \text{m}^3}
\]
This matches option d.