Question

if the right triangular prism is dilated using a scale factor of 3, what would be the volume of the dilated prism

b = 4cm, l = 7cm, h = 3cm

Answer 1

To find the volume of a right triangular prism, you first need to calculate the area of the triangular base and then multiply that area by the height of the prism.

1. **Calculate the area of the triangular base**:
The base of the triangular prism is a right triangle. The area \( A \) of a right triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, we can take the two legs of the triangle as the base and height:
- Base \( b = 4 \) cm
- Height \( h = 3 \) cm (using the same height as the prism's height is common).

So the area is:
\[
A = \frac{1}{2} \times 4 \, \text{cm} \times 3 \, \text{cm} = \frac{1}{2} \times 12 \, \text{cm}^2 = 6 \, \text{cm}^2
\]

2. **Calculate the volume of the prism**:
The volume \( V \) of the prism can be computed by multiplying the area of the base by the length:
\[
V = \text{base area} \times \text{length}
\]
Given the length \( l = 7 \) cm:
\[
V = 6 \, \text{cm}^2 \times 7 \, \text{cm} = 42 \, \text{cm}^3
\]

3. **Dilating the prism**:
When a prism is dilated with a scale factor \( k \), the volume of the new prism is given by:
\[
V' = k^3 \times V
\]
In this case, \( k = 3 \):
\[
V' = 3^3 \times 42 \, \text{cm}^3 = 27 \times 42 \, \text{cm}^3
\]

Now, calculate \( 27 \times 42 \):
\[
V' = 27 \times 42 = 1134 \, \text{cm}^3
\]

Thus, the volume of the dilated prism is **1134 cm³**.