To find the volume of a similar pyramid with a scale factor of \( \frac{2}{3} \), we can use the formula for the volume of a rectangular pyramid:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
First, we need to find the volume of the original pyramid before applying the scale factor.
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Determine the base area of the original pyramid.
The base is rectangular, so the area can be calculated as:\[ \text{Base Area} = \text{length} \times \text{width} = 24 \times 22 = 528 , \text{ft}^2 \]
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Determine the volume of the original pyramid using its height (28 ft):
\[ V_{\text{original}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times 528 \times 28 \]
\[ V_{\text{original}} = \frac{1}{3} \times 14784 = 4928 , \text{ft}^3 \]
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Use the scale factor to find the volume of the new pyramid:
The volume of a similar shape changes by the cube of the scaling factor:\[ V_{\text{new}} = V_{\text{original}} \times \left(\frac{2}{3}\right)^3 \]
\[ V_{\text{new}} = 4928 \times \frac{8}{27} = 4928 \times 0.2963 \approx 1460.15 , \text{ft}^3 \]
Thus, the volume of the new pyramid is approximately 1,460.15 ft³.
The correct response is: 1,460.15 ft³ (or 1,460.15 ft. cubed).
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