To find the volume of the original pyramid, we will first use the formula for the volume of a pyramid:
\[ V = \frac{l \times w \times h}{3} \]
From the information given:
- Height (h) = 15 feet
- Length (l) = 12 feet
- Width (w) = 6 feet
Now we can calculate the volume of the original pyramid:
\[ V = \frac{12 \times 6 \times 15}{3} \] \[ V = \frac{720}{3} \] \[ V = 240 \text{ ft}^3 \]
Now, when we dilate the pyramid with a scale factor of 2, all dimensions of the pyramid will be multiplied by 2. Therefore:
- New height = \(15 \times 2 = 30\) feet
- New length = \(12 \times 2 = 24\) feet
- New width = \(6 \times 2 = 12\) feet
Now we can recalculate the volume of the dilated pyramid:
\[ V' = \frac{l' \times w' \times h'}{3} \]
Where:
- New height (h') = 30 feet
- New length (l') = 24 feet
- New width (w') = 12 feet
Calculating the volume of the dilated pyramid:
\[ V' = \frac{24 \times 12 \times 30}{3} \] \[ V' = \frac{8640}{3} \] \[ V' = 2880 \text{ ft}^3 \]
Thus, the new volume of the pyramid after dilation is:
\[ \boxed{2880 \text{ ft}^3} \]
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