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 23
. What would be the volume of the new pyramid?
(1 point)
Responses
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
1,460.15 ft. cubed
182.52 ft.3
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
2,190.22 ft.3
2,190.22 ft. cubed
1,460.15 ft.3
1,460.15 ft. cubed
182.52 ft.3
Answers
GPT-4o mini
To find the volume of a rectangular pyramid, we can use the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
First, we need to calculate the volume of the original pyramid. The dimensions of the base are given as length \( l = 24 \) and width \( w = 22 \), and the height \( h = 28 \).
1. **Calculate the base area**:
\[
\text{Base Area} = l \times w = 24 \times 22 = 528 \, \text{ft}^2
\]
2. **Calculate the volume of the original pyramid**:
\[
V = \frac{1}{3} \times \text{Base Area} \times h = \frac{1}{3} \times 528 \times 28
\]
\[
V = \frac{1}{3} \times 14784 = 4928 \, \text{ft}^3
\]
Now, since the new pyramid is similar to the original pyramid with a scale factor of \( k = 23 \), the volume of the new pyramid can be calculated using the formula for similar solids:
\[
V' = k^3 \times V
\]
Substituting the values we have:
\[
V' = 23^3 \times 4928
\]
3. **Calculate \( 23^3 \)**:
\[
23^3 = 12167
\]
4. **Calculate the volume of the new pyramid**:
\[
V' = 12167 \times 4928
\]
\[
V' = 599,256 \, \text{ft}^3
\]
Based on the context and your provided answers, it appears you may have a typo or misunderstanding about the scale factor or its impact. The new volume, as calculated, is much larger than any provided answer options, which might imply a smaller scale factor or a miscalculation in the previous dimensions. Please verify the values and consider if the scale factor is instead \( \frac{1}{23} \) or a smaller number. This influences the outcome significantly.
If the scale factor is indeed \( \frac{1}{23} \):
Then:
\[
V' = \left( \frac{1}{23} \right)^3 \times 4928
\]
\[
V' = \frac{1}{12167} \times 4928 \approx 0.404 \, \text{ft}^3
\]
Please verify the parameters you provided for accuracy.
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
First, we need to calculate the volume of the original pyramid. The dimensions of the base are given as length \( l = 24 \) and width \( w = 22 \), and the height \( h = 28 \).
1. **Calculate the base area**:
\[
\text{Base Area} = l \times w = 24 \times 22 = 528 \, \text{ft}^2
\]
2. **Calculate the volume of the original pyramid**:
\[
V = \frac{1}{3} \times \text{Base Area} \times h = \frac{1}{3} \times 528 \times 28
\]
\[
V = \frac{1}{3} \times 14784 = 4928 \, \text{ft}^3
\]
Now, since the new pyramid is similar to the original pyramid with a scale factor of \( k = 23 \), the volume of the new pyramid can be calculated using the formula for similar solids:
\[
V' = k^3 \times V
\]
Substituting the values we have:
\[
V' = 23^3 \times 4928
\]
3. **Calculate \( 23^3 \)**:
\[
23^3 = 12167
\]
4. **Calculate the volume of the new pyramid**:
\[
V' = 12167 \times 4928
\]
\[
V' = 599,256 \, \text{ft}^3
\]
Based on the context and your provided answers, it appears you may have a typo or misunderstanding about the scale factor or its impact. The new volume, as calculated, is much larger than any provided answer options, which might imply a smaller scale factor or a miscalculation in the previous dimensions. Please verify the values and consider if the scale factor is instead \( \frac{1}{23} \) or a smaller number. This influences the outcome significantly.
If the scale factor is indeed \( \frac{1}{23} \):
Then:
\[
V' = \left( \frac{1}{23} \right)^3 \times 4928
\]
\[
V' = \frac{1}{12167} \times 4928 \approx 0.404 \, \text{ft}^3
\]
Please verify the parameters you provided for accuracy.