Question
Determine the volume of the rectangular prism if it were dilated using a scale factor of 4.
(1 point)
• 1, 536 m3
• 384 m3
• 6m3
• 96 m'
(1 point)
• 1, 536 m3
• 384 m3
• 6m3
• 96 m'
Answers
GPT-4o mini
To determine the volume of a rectangular prism after dilation with a scale factor of 4, we need to consider how dilation affects volume.
When a three-dimensional shape is dilated by a scale factor \( k \), the new volume is given by the formula:
\[
\text{New Volume} = k^3 \times \text{Original Volume}
\]
In this case, the scale factor \( k \) is 4. Therefore:
\[
k^3 = 4^3 = 64
\]
This means the new volume will be 64 times the original volume.
To find the original volume, we would need to either be given it directly or infer it from the list of possible answers. However, we don't have the original volume, but we can calculate what it might be if we look at the answer choices given, assuming they might represent the new volume after dilation.
If the new volume after dilation is one of the options provided, we can divide each option by 64 to see which one could potentially yield an integer volume for the original prism:
1. For **1,536 m³**:
\[
\frac{1,536}{64} = 24 \quad \text{(possible original volume)}
\]
2. For **384 m³**:
\[
\frac{384}{64} = 6 \quad \text{(possible original volume)}
\]
3. For **6 m³**:
\[
\frac{6}{64} = 0.09375 \quad \text{(not an integer)}
\]
4. For **96 m³**:
\[
\frac{96}{64} = 1.5 \quad \text{(not an integer)}
\]
From the calculations, the only potential new volumes that could correspond to integer original volumes are **1,536 m³** (which would correspond to an original volume of 24 m³) and **384 m³** (which would correspond to an original volume of 6 m³).
Since the question asks for the volume **after dilation**, we choose the largest option listed, which is:
**1,536 m³**.
When a three-dimensional shape is dilated by a scale factor \( k \), the new volume is given by the formula:
\[
\text{New Volume} = k^3 \times \text{Original Volume}
\]
In this case, the scale factor \( k \) is 4. Therefore:
\[
k^3 = 4^3 = 64
\]
This means the new volume will be 64 times the original volume.
To find the original volume, we would need to either be given it directly or infer it from the list of possible answers. However, we don't have the original volume, but we can calculate what it might be if we look at the answer choices given, assuming they might represent the new volume after dilation.
If the new volume after dilation is one of the options provided, we can divide each option by 64 to see which one could potentially yield an integer volume for the original prism:
1. For **1,536 m³**:
\[
\frac{1,536}{64} = 24 \quad \text{(possible original volume)}
\]
2. For **384 m³**:
\[
\frac{384}{64} = 6 \quad \text{(possible original volume)}
\]
3. For **6 m³**:
\[
\frac{6}{64} = 0.09375 \quad \text{(not an integer)}
\]
4. For **96 m³**:
\[
\frac{96}{64} = 1.5 \quad \text{(not an integer)}
\]
From the calculations, the only potential new volumes that could correspond to integer original volumes are **1,536 m³** (which would correspond to an original volume of 24 m³) and **384 m³** (which would correspond to an original volume of 6 m³).
Since the question asks for the volume **after dilation**, we choose the largest option listed, which is:
**1,536 m³**.