To calculate the new volume of a pyramid after applying a scale factor, you can follow these steps:
- Calculate the original volume of the pyramid using the formula \( V = \frac{1}{3} l w h \) where:
- \( l \) = length of the base,
- \( w \) = width of the base,
- \( h \) = height of the pyramid.
Given the dimensions:
- Length (l) = 12 ft
- Width (w) = 6 ft
- Height (h) = 15 ft
Plugging in the values: \[ V = \frac{1}{3} \times 12 , \text{ft} \times 6 , \text{ft} \times 15 , \text{ft} \]
Calculating it: \[ V = \frac{1}{3} \times 12 \times 6 \times 15 \] \[ V = \frac{1}{3} \times 1080 , \text{ft}^3 = 360 , \text{ft}^3 \]
- Apply the scale factor of dilation. The volume of a three-dimensional figure is affected by the cube of the scale factor. If the scale factor is \( k \), the new volume \( V' \) is given by: \[ V' = k^3 \times V \]
For a scale factor of 2: \[ k = 2 \quad \text{so} \quad k^3 = 2^3 = 8 \]
- Calculate the new volume: \[ V' = 8 \times 360 , \text{ft}^3 = 2880 , \text{ft}^3 \]
Thus, the new volume of the pyramid after dilation by a scale factor of 2 is \( \boxed{2880} , \text{ft}^3 \).