To find the volume of a rectangular pyramid, we can use the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
First, we need to calculate the volume of the original pyramid. The dimensions of the base are given as length \( l = 24 \) and width \( w = 22 \), and the height \( h = 28 \).
- Calculate the base area:
\[ \text{Base Area} = l \times w = 24 \times 22 = 528 , \text{ft}^2 \]
- Calculate the volume of the original pyramid:
\[ V = \frac{1}{3} \times \text{Base Area} \times h = \frac{1}{3} \times 528 \times 28 \] \[ V = \frac{1}{3} \times 14784 = 4928 , \text{ft}^3 \]
Now, since the new pyramid is similar to the original pyramid with a scale factor of \( k = 23 \), the volume of the new pyramid can be calculated using the formula for similar solids:
\[ V' = k^3 \times V \]
Substituting the values we have:
\[ V' = 23^3 \times 4928 \]
- Calculate \( 23^3 \):
\[ 23^3 = 12167 \]
- Calculate the volume of the new pyramid:
\[ V' = 12167 \times 4928 \] \[ V' = 599,256 , \text{ft}^3 \]
Based on the context and your provided answers, it appears you may have a typo or misunderstanding about the scale factor or its impact. The new volume, as calculated, is much larger than any provided answer options, which might imply a smaller scale factor or a miscalculation in the previous dimensions. Please verify the values and consider if the scale factor is instead \( \frac{1}{23} \) or a smaller number. This influences the outcome significantly.
If the scale factor is indeed \( \frac{1}{23} \):
Then:
\[ V' = \left( \frac{1}{23} \right)^3 \times 4928 \] \[ V' = \frac{1}{12167} \times 4928 \approx 0.404 , \text{ft}^3 \]
Please verify the parameters you provided for accuracy.