To find the volume of the pyramid with the same triangular base as the prism, we must first understand the relationship between the volumes of a prism and a pyramid.
The volume \( V \) of a prism is given by the formula:
\[ V = B \times h \]
where \( B \) is the area of the base and \( h \) is the height. The volume \( V \) of a pyramid is given by a similar formula:
\[ V = \frac{1}{3} B \times h \]
Given that the volume of the prism is \( 600 , \text{ft}^3 \), we can first find the area of the triangular base.
- Calculate the area (\( B \)) of the triangular base:
\[ B = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 20 , \text{ft} \times 6 , \text{ft} = 60 , \text{ft}^2 \]
- Use the volume of the prism to find the height (\( h \)):
We know:
\[ 600 , \text{ft}^3 = 60 , \text{ft}^2 \times h \]
So,
\[ h = \frac{600 , \text{ft}^3}{60 , \text{ft}^2} = 10 , \text{ft} \]
- Now, we can calculate the volume of the pyramid using the same base area and height:
\[ V_{\text{pyramid}} = \frac{1}{3} \times B \times h = \frac{1}{3} \times 60 , \text{ft}^2 \times 10 , \text{ft} = \frac{600 , \text{ft}^3}{3} = 200 , \text{ft}^3 \]
Thus, the volume of the pyramid is 200 ft.³.