To find the new volume of the pyramid after it has been dilated with a scale factor of 2, we first need to find the original volume.
The volume \( V \) of a pyramid is given by the formula:
\[ V = \frac{l \times w \times h}{3} \]
where:
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height.
Step 1: Calculate the original volume
Given:
- Length \( l = 12 , \text{ft} \)
- Width \( w = 6 , \text{ft} \)
- Height \( h = 15 , \text{ft} \)
Plugging these values into the formula:
\[ V = \frac{12 \times 6 \times 15}{3} \]
Calculate the product:
\[ 12 \times 6 = 72 \]
Then,
\[ 72 \times 15 = 1080 \]
Now, divide by 3:
\[ V = \frac{1080}{3} = 360 , \text{ft}^3 \]
Step 2: Calculate the new volume after dilation
When a three-dimensional shape is dilated by a scale factor \( k \), its volume is scaled by a factor of \( k^3 \). In this case, \( k = 2 \), so:
\[ \text{Scale factor for volume} = k^3 = 2^3 = 8 \]
Now, multiply the original volume by this scale factor:
\[ \text{New Volume} = 360 \times 8 = 2880 , \text{ft}^3 \]
Conclusion
After dilating the pyramid using a scale factor of 2, the new volume is:
\[ \boxed{2880 , \text{ft}^3} \]
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