Use the image to answer the question.

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

182.52 ft.3
182.25 ft. cubed

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

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

1,460.15 ft.3

Answer 1

To find the volume of the new pyramid, we first need to calculate the volume of the original pyramid and then use the scale factor to calculate the volume of the similar pyramid.

The formula for the volume \( V \) of a rectangular pyramid is:

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

1. Calculate the Base Area of the Original Pyramid:

   • The base dimensions are length = 24 ft and width = 22 ft.
   • The area \( A \) of the base is: \[ A = \text{length} \times \text{width} = 24 \times 22 = 528 \text{ ft}^2 \]

2. Calculate the Volume of the Original Pyramid:

   • The height of the pyramid is 28 ft.
   • The volume \( V \) is: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times 528 \times 28 \] \[ V = \frac{1}{3} \times 14784 = 4928 \text{ ft}^3 \]

3. Scale the Volume using the Scale Factor:

   • The scale factor given is \( \frac{2}{3} \).
   • The volume of a similar pyramid is scaled by the cube of the scale factor. Thus, the volume \( V' \) of the new pyramid is: \[ V' = V \times \left(\frac{2}{3}\right)^3 = 4928 \times \frac{8}{27} \] \[ V' = 4928 \times 0.296296 = 1460.15 \text{ ft}^3 \]

So, the volume of the new pyramid is 1,460.15 ft³.

Therefore, the correct response is:

1,460.15 ft.3