Question
Determine the volume of the rectangular prism if it were dilated using scale factor of 4. If the answer is 1,536 m^3 can you please explain how it is the answer
Answers
Answer
Oh I'm sorry the dimensions are 4 m by 3 m by 2 m
Answer
Thank you I understand now
Answered by
GPT-4o mini
To determine the volume of a rectangular prism after dilation, you need to know the original dimensions and how dilation affects volume.
1. **Understanding Dilation**: When you dilate a three-dimensional shape by a scale factor \( k \), each dimension (length, width, height) of the shape is multiplied by \( k \). Therefore, the formula for the new volume \( V' \) after dilation is:
\[
V' = k^3 \times V
\]
where \( V \) is the original volume of the prism.
2. **Given Scale Factor**: You mentioned a scale factor of 4. Therefore, \( k = 4 \), and we need to calculate \( k^3 \):
\[
k^3 = 4^3 = 64
\]
3. **Original Volume**: Next, you would need the original volume \( V \) of the prism. If the volume after dilation is given as \( 1,536 \, \text{m}^3 \), we can set up the equation:
\[
V' = 64 \times V
\]
So,
\[
1,536 = 64 \times V
\]
4. **Solve for \( V \)**:
To find the original volume \( V \):
\[
V = \frac{1,536}{64}
\]
Calculating this gives:
\[
V = 24 \, \text{m}^3
\]
5. **Conclusion**: Therefore, if the original volume of the rectangular prism is \( 24 \, \text{m}^3 \), and when dilated by a scale factor of 4, the volume becomes \( 1,536 \, \text{m}^3 \) as you mentioned.
In summary, if the volume of the rectangular prism after dilation is \( 1,536 \, \text{m}^3 \), it indicates that the original volume was \( 24 \, \text{m}^3 \) before being dilated by a scale factor of \( 4 \).
1. **Understanding Dilation**: When you dilate a three-dimensional shape by a scale factor \( k \), each dimension (length, width, height) of the shape is multiplied by \( k \). Therefore, the formula for the new volume \( V' \) after dilation is:
\[
V' = k^3 \times V
\]
where \( V \) is the original volume of the prism.
2. **Given Scale Factor**: You mentioned a scale factor of 4. Therefore, \( k = 4 \), and we need to calculate \( k^3 \):
\[
k^3 = 4^3 = 64
\]
3. **Original Volume**: Next, you would need the original volume \( V \) of the prism. If the volume after dilation is given as \( 1,536 \, \text{m}^3 \), we can set up the equation:
\[
V' = 64 \times V
\]
So,
\[
1,536 = 64 \times V
\]
4. **Solve for \( V \)**:
To find the original volume \( V \):
\[
V = \frac{1,536}{64}
\]
Calculating this gives:
\[
V = 24 \, \text{m}^3
\]
5. **Conclusion**: Therefore, if the original volume of the rectangular prism is \( 24 \, \text{m}^3 \), and when dilated by a scale factor of 4, the volume becomes \( 1,536 \, \text{m}^3 \) as you mentioned.
In summary, if the volume of the rectangular prism after dilation is \( 1,536 \, \text{m}^3 \), it indicates that the original volume was \( 24 \, \text{m}^3 \) before being dilated by a scale factor of \( 4 \).
Answered by
GPT-4o mini
No problem! Let's calculate the volume of the rectangular prism with the given dimensions \(4 \, \text{m}\), \(3 \, \text{m}\), and \(2 \, \text{m}\), and see what happens when we apply a dilation with a scale factor of \(4\).
### Step 1: Calculate the Original Volume
The volume \(V\) of a rectangular prism is calculated using the formula:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
Given the dimensions:
- Length = \(4 \, \text{m}\)
- Width = \(3 \, \text{m}\)
- Height = \(2 \, \text{m}\)
Now plug in the values:
\[
V = 4 \times 3 \times 2
\]
Calculating this:
\[
V = 12 \times 2 = 24 \, \text{m}^3
\]
### Step 2: Calculate the Volume After Dilation
Now, when we dilate the rectangular prism by a scale factor of \(4\), we will multiply each dimension by \(4\):
- New Length = \(4 \times 4 = 16 \, \text{m}\)
- New Width = \(3 \times 4 = 12 \, \text{m}\)
- New Height = \(2 \times 4 = 8 \, \text{m}\)
Now, calculate the new volume \(V'\):
\[
V' = \text{New Length} \times \text{New Width} \times \text{New Height}
\]
\[
V' = 16 \times 12 \times 8
\]
Calculating this step by step:
1. \(16 \times 12 = 192\)
2. \(192 \times 8 = 1536\)
So the volume after dilation is:
\[
V' = 1536 \, \text{m}^3
\]
### Conclusion
Thus, the volume of the rectangular prism after applying the dilation with a scale factor of \(4\) is indeed \(1536 \, \text{m}^3\). Thank you for providing the dimensions!
### Step 1: Calculate the Original Volume
The volume \(V\) of a rectangular prism is calculated using the formula:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
Given the dimensions:
- Length = \(4 \, \text{m}\)
- Width = \(3 \, \text{m}\)
- Height = \(2 \, \text{m}\)
Now plug in the values:
\[
V = 4 \times 3 \times 2
\]
Calculating this:
\[
V = 12 \times 2 = 24 \, \text{m}^3
\]
### Step 2: Calculate the Volume After Dilation
Now, when we dilate the rectangular prism by a scale factor of \(4\), we will multiply each dimension by \(4\):
- New Length = \(4 \times 4 = 16 \, \text{m}\)
- New Width = \(3 \times 4 = 12 \, \text{m}\)
- New Height = \(2 \times 4 = 8 \, \text{m}\)
Now, calculate the new volume \(V'\):
\[
V' = \text{New Length} \times \text{New Width} \times \text{New Height}
\]
\[
V' = 16 \times 12 \times 8
\]
Calculating this step by step:
1. \(16 \times 12 = 192\)
2. \(192 \times 8 = 1536\)
So the volume after dilation is:
\[
V' = 1536 \, \text{m}^3
\]
### Conclusion
Thus, the volume of the rectangular prism after applying the dilation with a scale factor of \(4\) is indeed \(1536 \, \text{m}^3\). Thank you for providing the dimensions!
Answered by
GPT-4o mini
You're welcome! I'm glad to hear that you understand now. If you have any more questions about geometry, volume, or any other topic, feel free to ask. Happy studying!
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