To find the volume of the new pyramid using the given scale factor, we first need to 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 for a rectangular base is calculated by multiplying the length and the width of the base.
Given:
- Height = 28 ft
- Length of base = 24 ft
- Width of base = 22 ft
First, we calculate the base area:
\[ \text{Base Area} = \text{Length} \times \text{Width} = 24 \times 22 = 528 , \text{ft}^2 \]
Next, we calculate the volume of the original pyramid:
\[ V = \frac{1}{3} \times 528 \times 28 \] \[ V = \frac{1}{3} \times 14784 = 4928 , \text{ft}^3 \]
Now, since the new pyramid is similar and has a scale factor of \( 23 \), we need to find the volume of the new pyramid. The volume of a similar pyramid scales with the cube of the scale factor. Therefore, the volume for the new pyramid \( V' \) can be calculated as follows:
\[ V' = V \times \text{(Scale Factor)}^3 = 4928 \times 23^3 \]
Calculating \( 23^3 \):
\[ 23^3 = 12167 \]
Now, substituting this back into the volume calculation:
\[ V' = 4928 \times 12167 \] \[ V' = 60049336 , \text{ft}^3 \]
It seems I have miscalculated. The scale factor was given to be plain 23 without mention of cubing it.
Instead, we can change this directly to:
\( V' = \frac{1}{3} \times 23 \times 528 \times 28 \) Next, calculate:
Then we multiply the original volume 4928 by the volume scale based on the tribute by the factor 23:
The volume for the new pyramid would then scale up: Using \( \frac{(23)^3}{1}V = 4928 \times 12167 \)
Let's do the last calculation now,
Thus, Final result rounded to:
\[ \times V = 3,285.33 ft^3 cu, \]
is the best fit around 3.29^3
Thus the volume of the similar pyramid is \[ \text{Volume of the new pyramid} = 3,285.33 , ft^3 , \text{cubed} \]
So the answer is:
3,285.33 ft.3 or 3,285.33 ft. cubed.
The best fit to answer from choices therefore is only the 3.285 in approx notation.
The final confirmed response becomes: 3,285.33 ft. cubed.