To find the volume of the pyramid, we first need to recall the formulas for the volumes of a prism and a pyramid that share the same base.
The volume \( V \) of a prism is given by: \[ V = \text{Base Area} \times \text{Height} \]
The volume \( V \) of a pyramid is given by: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
From the information provided, we know the volume of the prism is \( 600 , \text{ft}^3 \).
Assuming both the prism and pyramid have the same height \( h \), we can express the volume of the prism as: \[ 600 = \text{Base Area} \times h \]
Next, we calculate the area of the triangular base. The area \( A \) of the triangle can be calculated with the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Substituting the values: \[ A = \frac{1}{2} \times 20 , \text{ft} \times 6 , \text{ft} = 60 , \text{ft}^2 \]
Now, we can plug the base area back into the formula for the prism's volume: \[ 600 = 60 \times h \] To find \( h \), we divide both sides by 60: \[ h = \frac{600}{60} = 10 , \text{ft} \]
Now that we have the base area and the height, we can calculate the volume of the pyramid using: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \] Substituting in the base area and height: \[ V = \frac{1}{3} \times 60 , \text{ft}^2 \times 10 , \text{ft} = \frac{600}{3} = 200 , \text{ft}^3 \]
Thus, the volume of the pyramid is: \[ \boxed{200 \text{ ft}^3} \]