To find the volume of a similar rectangular pyramid with a given scale factor, we can use the following steps:
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Calculate the volume of the original pyramid. The formula for the volume \( V \) of a rectangular pyramid is:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
The base area of a rectangle is given by:
\[ \text{Base Area} = \text{length} \times \text{width} \]
In this case, the base dimensions are 28 and 22 (assuming these are the dimensions of the base rectangle), and the height is 24. Therefore,
\[ \text{Base Area} = 28 \times 22 = 616 , \text{ft}^2 \]
Now, we can find the volume:
\[ V = \frac{1}{3} \times 616 \times 24 = \frac{1}{3} \times 14784 = 4928 , \text{ft}^3 \]
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Scale the volume by the scale factor. The volume of a similar pyramid scales with the cube of the scale factor. If the scale factor is 23, then the scale factor for volume is \( 23^3 \).
First, calculate \( 23^3 \):
\[ 23^3 = 23 \times 23 \times 23 = 12167 \]
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Calculate the volume of the new pyramid.
\[ V_{\text{new}} = V_{\text{original}} \times 23^3 = 4928 \times 12167 \]
Now, performing the multiplication:
\[ V_{\text{new}} = 4928 \times 12167 \approx 60009936 , \text{ft}^3 \]
Thus, the final volume of the new pyramid will be \( 60009936 , \text{ft}^3 \). However, none of the responses provided match this calculated value. Please check the original dimensions or validate the information as necessary. If this information was intended to lead to one of the possible response values, an assumption about the dimensions being less than presented should be made, or an error may have occurred in communicating the query.
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