To find the volume of the pyramid after dilation by a scale factor of 2, we first need to calculate the volume of the original pyramid and then determine the volume of the dilated pyramid.
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Calculate the volume of the original pyramid: The formula for the volume of a pyramid is given by: \[ V = \frac{l \times w \times h}{3} \] where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
From the dimensions provided:
- Length (\( l \)) = 12 ft
- Width (\( w \)) = 6 ft
- Height (\( h \)) = 15 ft
Substitute the dimensions into the volume formula: \[ V = \frac{12 \times 6 \times 15}{3} \]
First, calculate \( 12 \times 6 \times 15 \): \[ 12 \times 6 = 72 \] \[ 72 \times 15 = 1080 \]
Now plug this back into the volume formula: \[ V = \frac{1080}{3} = 360 \text{ cubic feet} \]
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Calculate the new dimensions after dilation: Since we are dilating the pyramid by a scale factor of 2, we multiply each dimension by 2:
- New length (\( l' \)) = \( 12 \times 2 = 24 \) ft
- New width (\( w' \)) = \( 6 \times 2 = 12 \) ft
- New height (\( h' \)) = \( 15 \times 2 = 30 \) ft
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Calculate the volume of the dilated pyramid: Now we use the new dimensions to calculate the new volume \( V' \): \[ V' = \frac{l' \times w' \times h'}{3} = \frac{24 \times 12 \times 30}{3} \]
Calculate \( 24 \times 12 \times 30 \): \[ 24 \times 12 = 288 \] \[ 288 \times 30 = 8640 \]
Now plug this back into the volume formula: \[ V' = \frac{8640}{3} = 2880 \text{ cubic feet} \]
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Conclusion: The new volume of the pyramid after dilation is: \[ \boxed{2880} \text{ cubic feet} \]