Question

if the triangular prism B7m L12m H8m were dilated using a scale factor of 3, what would be the new volume?

Answer 1

To find the new volume of the triangular prism after dilating it with a scale factor of 3, we can follow these steps:

1. **Determine the volume of the original triangular prism:**
The volume \( V \) of a triangular prism can be calculated using the formula:

\[
V = \text{Base Area} \times \text{Height}
\]

First, calculate the area of the triangular base.

Given:
- Base \( B = 7 \, \text{m} \)
- Height of the triangular base \( H = 12 \, \text{m} \) (assuming this is the height of the triangle, not the height of the prism).

The area \( A \) of the triangular base can be computed as:

\[
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \, \text{m} \times 12 \, \text{m} = 42 \, \text{m}^2
\]

Now, use the height of the prism, \( H_{\text{prism}} = 8 \, \text{m} \):

\[
V = A \times H_{\text{prism}} = 42 \, \text{m}^2 \times 8 \, \text{m} = 336 \, \text{m}^3
\]

2. **Apply the scale factor of dilation:**
When an object is dilated using a scale factor \( k \), the new volume \( V' \) is given by:

\[
V' = k^3 \times V
\]

Here \( k = 3 \):

\[
V' = 3^3 \times 336 \, \text{m}^3 = 27 \times 336 \, \text{m}^3 = 9072 \, \text{m}^3
\]

Thus, the new volume of the dilated triangular prism is **9072 m³**.