Question
Suppose a similar rectangular pyramid to the one shown has a scale factor of 23 . What would be the volume of the new pyramid? (1 point) Responses 3,285.33 ft.3 3,285.33 ft. cubed 1,460.15 ft.3 1,460.15 ft. cubed 2,190.22 ft.3 2,190.22 ft. cubed 182.52 ft.3
height is 28
width is 24
length is 22
height is 28
width is 24
length is 22
Answers
GPT-4o mini
To find the volume of a rectangular pyramid, you can use the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
For a rectangular base, the base area \( A \) is calculated as:
\[
A = \text{Length} \times \text{Width}
\]
Given:
- Height = 28 ft
- Width = 24 ft
- Length = 22 ft
First, calculate the base area:
\[
A = 22 \, \text{ft} \times 24 \, \text{ft} = 528 \, \text{ft}^2
\]
Now, substitute the base area and the height into the volume formula:
\[
V = \frac{1}{3} \times 528 \, \text{ft}^2 \times 28 \, \text{ft}
\]
\[
V = \frac{1}{3} \times 14784 \, \text{ft}^3
\]
\[
V = 4928 \, \text{ft}^3
\]
Now that we have the volume of the original rectangular pyramid, we need to account for the scale factor of 23. The volume of a similar pyramid scales by the cube of the scale factor. Therefore, the volume \( V' \) of the new pyramid can be found as follows:
\[
V' = V \times \text{(Scale Factor)}^3
\]
\[
V' = 4928 \, \text{ft}^3 \times 23^3
\]
Calculating \( 23^3 \):
\[
23^3 = 12167
\]
Now, calculate the new volume:
\[
V' = 4928 \, \text{ft}^3 \times 12167 = 600,351,456 \, \text{ft}^3
\]
Thus, the volume of the new pyramid would be:
\[
600,351.456 \, \text{ft}^3
\]
Since your original options don't include this, please check to see if the scale factor or the dimensions given were correct. If they are, the calculation stands valid, and it may just be the options presented that are not matching.
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
For a rectangular base, the base area \( A \) is calculated as:
\[
A = \text{Length} \times \text{Width}
\]
Given:
- Height = 28 ft
- Width = 24 ft
- Length = 22 ft
First, calculate the base area:
\[
A = 22 \, \text{ft} \times 24 \, \text{ft} = 528 \, \text{ft}^2
\]
Now, substitute the base area and the height into the volume formula:
\[
V = \frac{1}{3} \times 528 \, \text{ft}^2 \times 28 \, \text{ft}
\]
\[
V = \frac{1}{3} \times 14784 \, \text{ft}^3
\]
\[
V = 4928 \, \text{ft}^3
\]
Now that we have the volume of the original rectangular pyramid, we need to account for the scale factor of 23. The volume of a similar pyramid scales by the cube of the scale factor. Therefore, the volume \( V' \) of the new pyramid can be found as follows:
\[
V' = V \times \text{(Scale Factor)}^3
\]
\[
V' = 4928 \, \text{ft}^3 \times 23^3
\]
Calculating \( 23^3 \):
\[
23^3 = 12167
\]
Now, calculate the new volume:
\[
V' = 4928 \, \text{ft}^3 \times 12167 = 600,351,456 \, \text{ft}^3
\]
Thus, the volume of the new pyramid would be:
\[
600,351.456 \, \text{ft}^3
\]
Since your original options don't include this, please check to see if the scale factor or the dimensions given were correct. If they are, the calculation stands valid, and it may just be the options presented that are not matching.