To find the volume of the new pyramid, we first need to calculate the volume of the original pyramid and then apply the scale factor.
The formula for the volume \( V \) of a rectangular pyramid is given by: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
The base area for this rectangular pyramid can be calculated by multiplying the length and width of the base: \[ \text{Base Area} = \text{Length} \times \text{Width} = 24 \times 22 = 528 \text{ ft}^2 \]
Now, substituting the base area and height into the volume formula: \[ V = \frac{1}{3} \times 528 \times 28 \] Calculating the volume: \[ V = \frac{1}{3} \times 528 \times 28 = \frac{1}{3} \times 14784 \approx 4928 \text{ ft}^3 \]
Now, to find the volume of the similar pyramid with a scale factor of 23, we use the fact that the volume of similar pyramids scales with the cube of the scale factor: \[ \text{Volume of new pyramid} = \text{Volume of original pyramid} \times (\text{scale factor})^3 \] \[ \text{Volume of new pyramid} = 4928 \times 23^3 \] Calculating \( 23^3 \): \[ 23^3 = 12167 \] Now, calculate the volume of the new pyramid: \[ \text{Volume of new pyramid} = 4928 \times 12167 \approx 599,758.56 \text{ ft}^3 \]
However, this number is much larger than the choices given. I must have made a mistake in understanding the scale factor since it seems like it shouldn't be 23. Let's check the calculations specifically for the volume of the original pyramid.
Now, let's verify all steps: If the height is \( 28 \), base lengths \( 24 \) and \( 22 \):
Original volume would give: \[ V = \frac{1}{3} \times 528 \times 28 = 4928 \text{ ft}^3 \]
Applying the scale cube: Though I suspect that the scale factor was miscommunicated as it seems incredibly massive.
However, if somehow meant that with:
Refeding it back down: If original volume is: V = \frac{1}{3} \times 528 \times 28 = \frac{1}{3} \times 14784 \approx 4928;
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