To determine the volume of the new pyramid with a scale factor of \( \frac{2}{3} \), we first need to calculate the volume of the original pyramid and then apply the scale factor to find the volume of the similar pyramid.
The formula for the volume \( V \) of a rectangular pyramid is given by:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
From your numbers (28, 24, and 22), we will assume they represent the length, width, and height of the original pyramid, respectively.
- Calculate the base area of the original pyramid:
\[ \text{Base Area} = \text{length} \times \text{width} = 28 , \text{ft} \times 24 , \text{ft} = 672 , \text{ft}^2 \]
- Calculate the volume of the original pyramid:
Using the height of 22 ft:
\[ V = \frac{1}{3} \times 672 , \text{ft}^2 \times 22 , \text{ft} = \frac{1}{3} \times 14784 , \text{ft}^3 = 4928 , \text{ft}^3 \]
- Calculate the volume of the new pyramid using the scale factor:
The volume of similar solids scales by the cube of the scale factor. Therefore, if the scale factor is \( \frac{2}{3} \), the new volume \( V' \) can be calculated as follows:
\[ V' = V \times \left( \frac{2}{3} \right)^3 = V \times \frac{8}{27} \]
Substituting the original volume:
\[ V' = 4928 , \text{ft}^3 \times \frac{8}{27} \approx 1460.15 , \text{ft}^3 \]
Thus, the volume of the new pyramid is approximately:
1,460.15 ft³
Therefore, the correct answer is:
c. 1,460.15 ft³
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