Question

If the rectangular prism shown were dilated by a scale factor of 12 . What would the volume of the dilated prism be? (1 point) Responses 120 cm2 120 cm2 15 cm2 15 cm2 60 cm2 60 cm2 30 cm2 30 cm2. 4cm 5cm 6 cm

Answer 1

To find the volume of a rectangular prism after it has been dilated by a scale factor, you first need to calculate the original volume of the prism and then apply the scale factor.

1. **Calculate the original volume** of the prism:
The volume \( V \) of a rectangular prism can be calculated using the formula:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
For the given dimensions \( 4 \, \text{cm} \), \( 5 \, \text{cm} \), and \( 6 \, \text{cm} \):
\[
V = 4 \times 5 \times 6 = 120 \, \text{cm}^3
\]

2. **Dilate the volume by the scale factor**:
When a prism is dilated by a scale factor of \( k \), the new volume \( V' \) is given by:
\[
V' = k^3 \times V
\]
In this case, \( k = 12 \):
\[
V' = 12^3 \times 120
\]

First, calculate \( 12^3 \):
\[
12^3 = 1728
\]

Now multiply by the original volume:
\[
V' = 1728 \times 120 = 207360 \, \text{cm}^3
\]

Therefore, the volume of the dilated prism is \( 207360 \, \text{cm}^3 \). It looks like none of the provided responses (120 cm², 15 cm², 60 cm², 30 cm²) match this calculation. Please verify if the response options given were correctly listed or clarify the question details.