Question
Armando found the volume of the figure below by using the steps shown.
A rectangular prism with a length of 18 inches, width of 5 inches, and height of 8 inches. A triangular prism with a base of 18 inches and height of 6 inches. The prism has a height of 5 inches.
Total Volume = Volume of rectangular prism + volume of triangular prism = 5 (8) (18) + one-half (6) (18) (8) = 720 + 432 = 1,152 cubic inches.
What mistake, if any, did Armando make?
Armando determined the volume correctly.
Armando used the wrong formula for the total volume.
Armando used the wrong dimensions for the triangular prism.
Armando did not apply the order of operations correctly.
A rectangular prism with a length of 18 inches, width of 5 inches, and height of 8 inches. A triangular prism with a base of 18 inches and height of 6 inches. The prism has a height of 5 inches.
Total Volume = Volume of rectangular prism + volume of triangular prism = 5 (8) (18) + one-half (6) (18) (8) = 720 + 432 = 1,152 cubic inches.
What mistake, if any, did Armando make?
Armando determined the volume correctly.
Armando used the wrong formula for the total volume.
Armando used the wrong dimensions for the triangular prism.
Armando did not apply the order of operations correctly.
Answers
Answered by
GPT-4o mini
Let's break down the calculation to identify any mistakes Armando may have made.
1. **Volume of the Rectangular Prism**:
The formula for the volume of a rectangular prism is:
\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]
Using the dimensions given:
\[
\text{Volume} = 18 \, \text{inches} \times 5 \, \text{inches} \times 8 \, \text{inches} = 720 \, \text{cubic inches}
\]
This part is calculated correctly.
2. **Volume of the Triangular Prism**:
The formula for the volume of a triangular prism is:
\[
\text{Volume} = \text{Base Area} \times \text{height}
\]
The base area of the triangle can be found using the formula:
\[
\text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the triangular prism, the base is 18 inches and the height is 6 inches. Therefore:
\[
\text{Base Area} = \frac{1}{2} \times 18 \, \text{inches} \times 6 \, \text{inches} = 54 \, \text{square inches}
\]
Now, we multiply the base area by the height of the prism (which is 5 inches):
\[
\text{Volume} = \text{Base Area} \times \text{height} = 54 \, \text{square inches} \times 5 \, \text{inches} = 270 \, \text{cubic inches}
\]
3. **Total Volume**:
Now, we add the volume of the rectangular prism and the volume of the triangular prism:
\[
\text{Total Volume} = 720 \, \text{cubic inches} + 270 \, \text{cubic inches} = 990 \, \text{cubic inches}
\]
Armando originally calculated the volume of the triangular prism incorrectly. Instead of \(432 \, \text{cubic inches}\), it should be \(270 \, \text{cubic inches}\). Therefore, the total volume he calculated is also incorrect.
So the correct answer is:
**Armando used the wrong dimensions for the triangular prism.**
1. **Volume of the Rectangular Prism**:
The formula for the volume of a rectangular prism is:
\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]
Using the dimensions given:
\[
\text{Volume} = 18 \, \text{inches} \times 5 \, \text{inches} \times 8 \, \text{inches} = 720 \, \text{cubic inches}
\]
This part is calculated correctly.
2. **Volume of the Triangular Prism**:
The formula for the volume of a triangular prism is:
\[
\text{Volume} = \text{Base Area} \times \text{height}
\]
The base area of the triangle can be found using the formula:
\[
\text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the triangular prism, the base is 18 inches and the height is 6 inches. Therefore:
\[
\text{Base Area} = \frac{1}{2} \times 18 \, \text{inches} \times 6 \, \text{inches} = 54 \, \text{square inches}
\]
Now, we multiply the base area by the height of the prism (which is 5 inches):
\[
\text{Volume} = \text{Base Area} \times \text{height} = 54 \, \text{square inches} \times 5 \, \text{inches} = 270 \, \text{cubic inches}
\]
3. **Total Volume**:
Now, we add the volume of the rectangular prism and the volume of the triangular prism:
\[
\text{Total Volume} = 720 \, \text{cubic inches} + 270 \, \text{cubic inches} = 990 \, \text{cubic inches}
\]
Armando originally calculated the volume of the triangular prism incorrectly. Instead of \(432 \, \text{cubic inches}\), it should be \(270 \, \text{cubic inches}\). Therefore, the total volume he calculated is also incorrect.
So the correct answer is:
**Armando used the wrong dimensions for the triangular prism.**