To find the volume of the new pyramid after applying the scale factor, we first need to calculate the volume of the original pyramid.
The formula for the volume \( V \) of a rectangular pyramid is given by:
\[ V = \frac{1}{3} \times L \times W \times H \]
where:
- \( L \) is the length,
- \( W \) is the width,
- \( H \) is the height.
Given:
- \( L = 22 \) ft,
- \( W = 24 \) ft,
- \( H = 28 \) ft.
Plugging in the values, we calculate the volume of the original pyramid:
\[ V = \frac{1}{3} \times 22 \times 24 \times 28 \]
Calculating step by step:
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Calculate the product of the dimensions: \[ 22 \times 24 = 528 \] \[ 528 \times 28 = 14784 \]
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Now, divide by 3: \[ V = \frac{14784}{3} = 4928 \text{ ft}^3 \]
Now that we have the volume of the original pyramid, we can find the volume of the new pyramid that is similar and scaled down by the scale factor of \( \frac{2}{3} \).
The volume of a similar pyramid scales by the cube of the scale factor. Therefore, the volume of the new pyramid \( V' \) is given by:
\[ V' = V \times \left( \frac{2}{3} \right)^3 \]
Calculating \( \left( \frac{2}{3} \right)^3 \):
\[ \left( \frac{2}{3} \right)^3 = \frac{8}{27} \]
Now calculating the volume of the new pyramid:
\[ V' = 4928 \times \frac{8}{27} \]
Calculating \( 4928 \times 8 \):
\[ 4928 \times 8 = 39424 \]
Now dividing by 27:
\[ V' = \frac{39424}{27} \approx 1457.56 \text{ ft}^3 \]
Therefore, the volume of the new pyramid is approximately 1457.56 ft³.
Comparing this with the responses provided, it's approximately equal to 1,460.15 ft³.
Thus, the answer is: 1,460.15 ft³.
1 answer