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 times 2
b = 4cm, l = 7cm, h = 3cm times 2
Answers
GPT-4o mini
To find the volume of the right triangular prism after dilation, we first need to calculate the original volume of the prism and how dilation affects the volume.
1. **Calculate the volume of the original prism:**
The formula for the volume \( V \) of a prism is given by:
\[
V = \text{Base Area} \times \text{Height}
\]
The base of the triangular prism can be calculated using the dimensions provided. Since it is a right triangular prism, the base area \( A \) of the triangle can be calculated as:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, we'll assume \( b = 4 \, \text{cm} \) is the base of the triangle and \( h = 3 \, \text{cm} \) is the height of the triangle. For the triangular base:
\[
A = \frac{1}{2} \times 4 \times 3 = 6 \, \text{cm}^2
\]
Given the length \( l \) of the prism is \( 7 \, \text{cm} \), now we can compute the volume of the prism:
\[
V = A \times l = 6 \times 7 = 42 \, \text{cm}^3
\]
2. **Effect of dilation on volume:**
When a three-dimensional object is dilated by a scale factor \( k \), the volume of the object changes by a factor of \( k^3 \). Here, the scale factor is 3.
\[
\text{New Volume} = \text{Original Volume} \times \text{Scale Factor}^3
\]
\[
= 42 \, \text{cm}^3 \times 3^3
\]
\[
= 42 \, \text{cm}^3 \times 27
\]
\[
= 1134 \, \text{cm}^3
\]
Therefore, the volume of the dilated prism is **1134 cm³**.
1. **Calculate the volume of the original prism:**
The formula for the volume \( V \) of a prism is given by:
\[
V = \text{Base Area} \times \text{Height}
\]
The base of the triangular prism can be calculated using the dimensions provided. Since it is a right triangular prism, the base area \( A \) of the triangle can be calculated as:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, we'll assume \( b = 4 \, \text{cm} \) is the base of the triangle and \( h = 3 \, \text{cm} \) is the height of the triangle. For the triangular base:
\[
A = \frac{1}{2} \times 4 \times 3 = 6 \, \text{cm}^2
\]
Given the length \( l \) of the prism is \( 7 \, \text{cm} \), now we can compute the volume of the prism:
\[
V = A \times l = 6 \times 7 = 42 \, \text{cm}^3
\]
2. **Effect of dilation on volume:**
When a three-dimensional object is dilated by a scale factor \( k \), the volume of the object changes by a factor of \( k^3 \). Here, the scale factor is 3.
\[
\text{New Volume} = \text{Original Volume} \times \text{Scale Factor}^3
\]
\[
= 42 \, \text{cm}^3 \times 3^3
\]
\[
= 42 \, \text{cm}^3 \times 27
\]
\[
= 1134 \, \text{cm}^3
\]
Therefore, the volume of the dilated prism is **1134 cm³**.