Question

Question
Use the image to answer the question.

A 4-sided pyramid is labeled with a height from the top to base of 28, length of base of 24, and width of base of 22.

Suppose a similar rectangular pyramid to the one shown has a scale factor of 2/3
. What would be the volume of the new pyramid?

(1 point)
Responses which answer is correct and why

2,190.22 ft.3
2,190.22 ft. cubed

1,460.15 ft.3
1,460.15 ft. cubed

3,285.33 ft.3
3,285.33 ft. cubed

182.52 ft.3
182.25 ft. cubed
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Answer 1

To find the volume of a similar pyramid with a scale factor of \( \frac{2}{3} \), we can use the formula for the volume of a rectangular pyramid:

\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]

First, we need to find the volume of the original pyramid before applying the scale factor.

1. **Determine the base area of the original pyramid.**
The base is rectangular, so the area can be calculated as:

\[
\text{Base Area} = \text{length} \times \text{width} = 24 \times 22 = 528 \, \text{ft}^2
\]

2. **Determine the volume of the original pyramid using its height (28 ft):**

\[
V_{\text{original}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times 528 \times 28
\]

\[
V_{\text{original}} = \frac{1}{3} \times 14784 = 4928 \, \text{ft}^3
\]

3. **Use the scale factor to find the volume of the new pyramid:**
The volume of a similar shape changes by the cube of the scaling factor:

\[
V_{\text{new}} = V_{\text{original}} \times \left(\frac{2}{3}\right)^3
\]

\[
V_{\text{new}} = 4928 \times \frac{8}{27} = 4928 \times 0.2963 \approx 1460.15 \, \text{ft}^3
\]

Thus, the volume of the new pyramid is approximately **1,460.15 ft³**.

The correct response is:
**1,460.15 ft³** (or **1,460.15 ft. cubed**).