Question

Determine the volume of the rectangular prism if it were dilated using a scale factor of 4.
the length is 4 m the width is 3 m the height is 2 m

Answer 1

To determine the volume of the rectangular prism after it has been dilated using a scale factor of 4, we can follow these steps:

1. **Calculate the original volume of the rectangular prism** using the formula for volume \( V = \text{length} \times \text{width} \times \text{height} \).

Given:
- Length = 4 m
- Width = 3 m
- Height = 2 m

\[
V = 4 \, \text{m} \times 3 \, \text{m} \times 2 \, \text{m} = 24 \, \text{m}^3
\]

2. **Apply the scale factor** to each dimension of the prism:
- New Length = \( 4 \times 4 = 16 \, \text{m} \)
- New Width = \( 4 \times 3 = 12 \, \text{m} \)
- New Height = \( 4 \times 2 = 8 \, \text{m} \)

3. **Calculate the new volume** of the dilated prism using the same volume formula:

\[
V_{\text{new}} = \text{New Length} \times \text{New Width} \times \text{New Height}
\]
\[
V_{\text{new}} = 16 \, \text{m} \times 12 \, \text{m} \times 8 \, \text{m}
\]
\[
V_{\text{new}} = 1536 \, \text{m}^3
\]

Thus, the volume of the rectangular prism after dilation with a scale factor of 4 is **1536 m³**.

Alternatively, since volume scales by the cube of the scale factor, we could also compute it as:

\[
V_{\text{dilated}} = \text{Original Volume} \times (\text{scale factor})^3
\]
\[
V_{\text{dilated}} = 24 \, \text{m}^3 \times 4^3 = 24 \, \text{m}^3 \times 64 = 1536 \, \text{m}^3
\]

This confirms the new volume is indeed **1536 m³**.