To find the volume of the original pyramid with dimensions 15 ft (length), 12 ft (width), and 6 ft (height), we can use the formula for the volume of a pyramid:
\[ V = \frac{l \cdot w \cdot h}{3} \]
Substituting the given dimensions into the formula:
\[ V = \frac{15 , \text{ft} \cdot 12 , \text{ft} \cdot 6 , \text{ft}}{3} \]
Calculating the product of the dimensions:
\[ V = \frac{15 \cdot 12 \cdot 6}{3} \]
Calculating \(15 \cdot 12\):
\[ 15 \cdot 12 = 180 \]
Now calculating \(180 \cdot 6\):
\[ 180 \cdot 6 = 1080 \]
Now dividing by 3:
\[ V = \frac{1080}{3} = 360 , \text{ft}^3 \]
The volume of the original pyramid is \(360 , \text{ft}^3\).
Now, when the pyramid is dilated using a scale factor of 2, the new dimensions will be:
\[ \text{New length} = 15 \cdot 2 = 30 , \text{ft} \] \[ \text{New width} = 12 \cdot 2 = 24 , \text{ft} \] \[ \text{New height} = 6 \cdot 2 = 12 , \text{ft} \]
Now, we can find the volume of the dilated pyramid using the same formula:
\[ V_{\text{new}} = \frac{l_{\text{new}} \cdot w_{\text{new}} \cdot h_{\text{new}}}{3} \]
Substituting the new dimensions into the formula:
\[ V_{\text{new}} = \frac{30 , \text{ft} \cdot 24 , \text{ft} \cdot 12 , \text{ft}}{3} \]
Calculating the product of the new dimensions:
\[ V_{\text{new}} = \frac{30 \cdot 24 \cdot 12}{3} \]
Calculating \(30 \cdot 24\):
\[ 30 \cdot 24 = 720 \]
Now calculating \(720 \cdot 12\):
\[ 720 \cdot 12 = 8640 \]
Now dividing by 3:
\[ V_{\text{new}} = \frac{8640}{3} = 2880 , \text{ft}^3 \]
Thus, the volume of the dilated pyramid is \(\boxed{2880 , \text{ft}^3}\).