To find the volume of a pyramid, the formula is:
\[ V = \frac{1}{3} \times l \times w \times h \]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height of the pyramid.
Given your initial dimensions:
- Height (\( h \)) = 15 ft
- Width (\( w \)) = 12 ft
- Length (\( l \)) = 6 ft
First, calculate the original volume:
\[ V = \frac{1}{3} \times 6 \times 12 \times 15 \]
Calculating this step-by-step:
\[ 6 \times 12 = 72 \] \[ 72 \times 15 = 1080 \] \[ V = \frac{1}{3} \times 1080 = 360 \text{ ft}^3 \]
Now, if the pyramid is dilated using a scale factor of 2, all dimensions will be multiplied by this scale factor. The new dimensions will be:
- New Height = \( 15 \times 2 = 30 \) ft
- New Width = \( 12 \times 2 = 24 \) ft
- New Length = \( 6 \times 2 = 12 \) ft
Next, calculate the new volume using the new dimensions:
\[ V' = \frac{1}{3} \times 12 \times 24 \times 30 \]
Calculating this step-by-step:
\[ 12 \times 24 = 288 \] \[ 288 \times 30 = 8640 \] \[ V' = \frac{1}{3} \times 8640 = 2880 \text{ ft}^3 \]
So, the new volume after the dilation is \( \text{2880 ft}^3 \).