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.
-
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:
\[ V = A \times l = 6 \times 7 = 42 , \text{cm}^3 \]
-
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 answer