Question
Sophie found the volume of the figure below by using the steps shown.
A rectangular prism with a length of 28 millimeters, width of 21 millimeters, and height of 4 millimeters. A rectangular pyramid with a base of 28 millimeters by 21 millimeters and height of 26 millimeters.
Total Volume = B h + one-third B h = 4 (21) (28) + one-third (21) (28) (30) = 2,352 + 5,880 = 8,232 cubic millimeters.
What mistake, if any, did Sophie make?
Sophie determined the volume correctly.
Sophie used the wrong formula for the total volume.
Sophie used the wrong height for the pyramid.
Sophie did not apply the order of operations correctly.
A rectangular prism with a length of 28 millimeters, width of 21 millimeters, and height of 4 millimeters. A rectangular pyramid with a base of 28 millimeters by 21 millimeters and height of 26 millimeters.
Total Volume = B h + one-third B h = 4 (21) (28) + one-third (21) (28) (30) = 2,352 + 5,880 = 8,232 cubic millimeters.
What mistake, if any, did Sophie make?
Sophie determined the volume correctly.
Sophie used the wrong formula for the total volume.
Sophie used the wrong height for the pyramid.
Sophie did not apply the order of operations correctly.
Answers
Answered by
GPT-4o mini
Sophie made a mistake in the way she calculated the volume of the rectangular pyramid.
1. **Checking the dimensions:** The base area (B) of the pyramid with base dimensions of 28 mm by 21 mm is indeed calculated as \(28 \times 21 = 588\) square millimeters.
2. **Height of the pyramid:** Sophie used a height of 30 mm for the pyramid in the formula for volume, instead of the correct height of 26 mm.
The formula for the volume of the rectangular pyramid should be:
\[
\text{Volume of Pyramid} = \frac{1}{3} B h = \frac{1}{3} \times 588 \times 26
\]
3. **Calculating the correct volume of the pyramid:** Now calculate:
\[
\frac{1}{3} \times 588 \times 26 = \frac{1}{3} \times 15288 = 5096 \text{ cubic millimeters (correct volume for pyramid)}
\]
4. **Total volume:** Now, adding the volume of the rectangular prism:
\[
\text{Volume of Prism} = 28 \times 21 \times 4 = 2352 \text{ cubic millimeters}
\]
Thus, the total volume would be:
\[
\text{Total Volume} = 2352 + 5096 = 7448 \text{ cubic millimeters}
\]
Overall, the mistake was in using the incorrect height for the pyramid. The correct answer is:
**Sophie used the wrong height for the pyramid.**
1. **Checking the dimensions:** The base area (B) of the pyramid with base dimensions of 28 mm by 21 mm is indeed calculated as \(28 \times 21 = 588\) square millimeters.
2. **Height of the pyramid:** Sophie used a height of 30 mm for the pyramid in the formula for volume, instead of the correct height of 26 mm.
The formula for the volume of the rectangular pyramid should be:
\[
\text{Volume of Pyramid} = \frac{1}{3} B h = \frac{1}{3} \times 588 \times 26
\]
3. **Calculating the correct volume of the pyramid:** Now calculate:
\[
\frac{1}{3} \times 588 \times 26 = \frac{1}{3} \times 15288 = 5096 \text{ cubic millimeters (correct volume for pyramid)}
\]
4. **Total volume:** Now, adding the volume of the rectangular prism:
\[
\text{Volume of Prism} = 28 \times 21 \times 4 = 2352 \text{ cubic millimeters}
\]
Thus, the total volume would be:
\[
\text{Total Volume} = 2352 + 5096 = 7448 \text{ cubic millimeters}
\]
Overall, the mistake was in using the incorrect height for the pyramid. The correct answer is:
**Sophie used the wrong height for the pyramid.**