Question
If the triangular prism were dilated using a scale factor of 3, what would be the new volume?
the length is 12 m
the width is 7 m
the height is 8 m
the length is 12 m
the width is 7 m
the height is 8 m
Answers
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Answered by
GPT-4o mini
To find the volume of a triangular prism, you can use the formula:
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
The base area for a triangular prism can be calculated using the formula for the area of a triangle:
\[
\text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, let's assume the "width" is the base of the triangle and the "height" is the height of the triangle.
Given:
- Length of the prism = 12 m
- Width (base of the triangle) = 7 m
- Height of the triangle = 8 m
First, calculate the area of the triangular base:
\[
\text{Base Area} = \frac{1}{2} \times 7 \, \text{m} \times 8 \, \text{m} = \frac{1}{2} \times 56 \, \text{m}^2 = 28 \, \text{m}^2
\]
Next, calculate the volume of the prism using the length:
\[
\text{Volume} = \text{Base Area} \times \text{Length} = 28 \, \text{m}^2 \times 12 \, \text{m} = 336 \, \text{m}^3
\]
When the prism is dilated using a scale factor of 3, each dimension (length, width, height) is multiplied by 3. Therefore, the new dimensions will be:
- New length = \( 12 \, \text{m} \times 3 = 36 \, \text{m} \)
- New width (base of triangle) = \( 7 \, \text{m} \times 3 = 21 \, \text{m} \)
- New height of the triangle = \( 8 \, \text{m} \times 3 = 24 \, \text{m} \)
Now, we calculate the new base area:
\[
\text{New Base Area} = \frac{1}{2} \times 21 \, \text{m} \times 24 \, \text{m} = \frac{1}{2} \times 504 \, \text{m}^2 = 252 \, \text{m}^2
\]
Now calculate the new volume of the prism:
\[
\text{New Volume} = \text{New Base Area} \times \text{New Length} = 252 \, \text{m}^2 \times 36 \, \text{m} = 9072 \, \text{m}^3
\]
Thus, the new volume of the dilated triangular prism is:
\[
\boxed{9072 \, \text{m}^3}
\]
Alternatively, since volume scales by the cube of the scale factor, you can also calculate it simply by multiplying the original volume by \( 3^3 \):
\[
\text{New Volume} = 336 \, \text{m}^3 \times 27 = 9072 \, \text{m}^3
\]
This confirms our previous calculation.
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
The base area for a triangular prism can be calculated using the formula for the area of a triangle:
\[
\text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, let's assume the "width" is the base of the triangle and the "height" is the height of the triangle.
Given:
- Length of the prism = 12 m
- Width (base of the triangle) = 7 m
- Height of the triangle = 8 m
First, calculate the area of the triangular base:
\[
\text{Base Area} = \frac{1}{2} \times 7 \, \text{m} \times 8 \, \text{m} = \frac{1}{2} \times 56 \, \text{m}^2 = 28 \, \text{m}^2
\]
Next, calculate the volume of the prism using the length:
\[
\text{Volume} = \text{Base Area} \times \text{Length} = 28 \, \text{m}^2 \times 12 \, \text{m} = 336 \, \text{m}^3
\]
When the prism is dilated using a scale factor of 3, each dimension (length, width, height) is multiplied by 3. Therefore, the new dimensions will be:
- New length = \( 12 \, \text{m} \times 3 = 36 \, \text{m} \)
- New width (base of triangle) = \( 7 \, \text{m} \times 3 = 21 \, \text{m} \)
- New height of the triangle = \( 8 \, \text{m} \times 3 = 24 \, \text{m} \)
Now, we calculate the new base area:
\[
\text{New Base Area} = \frac{1}{2} \times 21 \, \text{m} \times 24 \, \text{m} = \frac{1}{2} \times 504 \, \text{m}^2 = 252 \, \text{m}^2
\]
Now calculate the new volume of the prism:
\[
\text{New Volume} = \text{New Base Area} \times \text{New Length} = 252 \, \text{m}^2 \times 36 \, \text{m} = 9072 \, \text{m}^3
\]
Thus, the new volume of the dilated triangular prism is:
\[
\boxed{9072 \, \text{m}^3}
\]
Alternatively, since volume scales by the cube of the scale factor, you can also calculate it simply by multiplying the original volume by \( 3^3 \):
\[
\text{New Volume} = 336 \, \text{m}^3 \times 27 = 9072 \, \text{m}^3
\]
This confirms our previous calculation.
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