To find the new volume after dilating the pyramid, we first need to calculate the volume of the original pyramid using the formula for the volume of a pyramid:
\[ V = \frac{1}{3} l w h \]
where:
- \( l \) is the length of the base,
- \( w \) is the width of the base,
- \( h \) is the height of the pyramid.
Given:
- Length \( l = 12 \) feet,
- Width \( w = 6 \) feet,
- Height \( h = 15 \) feet.
Now we can calculate the original volume:
\[ V = \frac{1}{3} \times 12 \times 6 \times 15 \] \[ V = \frac{1}{3} \times 720 \] \[ V = 240 \text{ cubic feet} \]
Next, when the pyramid is dilated by a scale factor of 2, the dimensions of the pyramid will change as follows:
- New length = \( 2 \times 12 = 24 \) feet,
- New width = \( 2 \times 6 = 12 \) feet,
- New height = \( 2 \times 15 = 30 \) feet.
Now we can calculate the volume of the dilated pyramid:
\[ V' = \frac{1}{3} \times 24 \times 12 \times 30 \] \[ V' = \frac{1}{3} \times 8640 \] \[ V' = 2880 \text{ cubic feet} \]
Thus, the new volume of the dilated pyramid is \( \boxed{2880} \) cubic feet.
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