Question
Which figure has a greater volume, and how much greater is it?
2 figures. One figure has a rectangular prism with length of 36 feet, width of 21 feet, and height of 18 feet. The rectangular pyramid has a base of 36 feet by 21 feet, and height of 9 feet. The other figure has a rectangular prism with length of 36 feet, width of 21 feet, and height of 18 feet. A triangular prism has a triangular base with base of 36 feet and height of 9 feet. The prism has a height of 21 feet.
The volume of figure 1 is 972 feet cubed greater than the volume of figure 2.
The volume of figure 2 is 972 feet cubed greater than the volume of figure 1.
The volume of figure 1 is 1,134 feet cubed greater than the volume of figure 2.
The volume of figure 2 is 1,134 feet cubed greater than the volume of figure 1.
2 figures. One figure has a rectangular prism with length of 36 feet, width of 21 feet, and height of 18 feet. The rectangular pyramid has a base of 36 feet by 21 feet, and height of 9 feet. The other figure has a rectangular prism with length of 36 feet, width of 21 feet, and height of 18 feet. A triangular prism has a triangular base with base of 36 feet and height of 9 feet. The prism has a height of 21 feet.
The volume of figure 1 is 972 feet cubed greater than the volume of figure 2.
The volume of figure 2 is 972 feet cubed greater than the volume of figure 1.
The volume of figure 1 is 1,134 feet cubed greater than the volume of figure 2.
The volume of figure 2 is 1,134 feet cubed greater than the volume of figure 1.
Answers
GPT-4o mini
To solve the problem, we need to calculate the volumes of both figures and compare them.
### Figure 1:
1. **Rectangular Prism Volume**:
- Length = 36 feet
- Width = 21 feet
- Height = 18 feet
- Volume = length × width × height = \(36 \times 21 \times 18\)
\[
Volume_{prism} = 36 \times 21 \times 18 = 13608 \text{ cubic feet}
\]
2. **Rectangular Pyramid Volume**:
- Base Length = 36 feet
- Base Width = 21 feet
- Height = 9 feet
- Volume = \(\frac{1}{3} \times \) base area \(\times\) height = \(\frac{1}{3} \times (36 \times 21) \times 9\)
\[
Volume_{pyramid} = \frac{1}{3} \times (36 \times 21) \times 9 = \frac{1}{3} \times 756 \times 9 = \frac{6804}{3} = 2268 \text{ cubic feet}
\]
3. **Total Volume of Figure 1**:
\[
Volume_{figure1} = Volume_{prism} + Volume_{pyramid} = 13608 + 2268 = 15876 \text{ cubic feet}
\]
### Figure 2:
1. **Rectangular Prism Volume** (same as Figure 1):
\[
Volume_{prism} = 13608 \text{ cubic feet}
\]
2. **Triangular Prism Volume**:
- Base = 36 feet
- Height of the triangle = 9 feet
- Height of the prism = 21 feet
- Volume = \(\frac{1}{2} \times \) base \(\times\) height of triangle \(\times\) height of prism = \(\frac{1}{2} \times 36 \times 9 \times 21\)
\[
Volume_{triangular\_prism} = \frac{1}{2} \times 36 \times 9 \times 21 = \frac{1}{2} \times 36 \times 189 = 36 \times 94.5 = 3402 \text{ cubic feet}
\]
3. **Total Volume of Figure 2**:
\[
Volume_{figure2} = Volume_{prism} + Volume_{triangular\_prism} = 13608 + 3402 = 17010 \text{ cubic feet}
\]
### Comparing the volumes of the figures:
- \(Volume_{figure1} = 15876\) cubic feet
- \(Volume_{figure2} = 17010\) cubic feet
\[
Volume_{figure2} - Volume_{figure1} = 17010 - 15876 = 1344 \text{ cubic feet}
\]
### Conclusion:
The volume of Figure 2 is greater than the volume of Figure 1, and it is 1344 cubic feet greater. Thus, the correct statement is:
**The volume of figure 2 is 1,344 feet cubed greater than the volume of figure 1.**
### Figure 1:
1. **Rectangular Prism Volume**:
- Length = 36 feet
- Width = 21 feet
- Height = 18 feet
- Volume = length × width × height = \(36 \times 21 \times 18\)
\[
Volume_{prism} = 36 \times 21 \times 18 = 13608 \text{ cubic feet}
\]
2. **Rectangular Pyramid Volume**:
- Base Length = 36 feet
- Base Width = 21 feet
- Height = 9 feet
- Volume = \(\frac{1}{3} \times \) base area \(\times\) height = \(\frac{1}{3} \times (36 \times 21) \times 9\)
\[
Volume_{pyramid} = \frac{1}{3} \times (36 \times 21) \times 9 = \frac{1}{3} \times 756 \times 9 = \frac{6804}{3} = 2268 \text{ cubic feet}
\]
3. **Total Volume of Figure 1**:
\[
Volume_{figure1} = Volume_{prism} + Volume_{pyramid} = 13608 + 2268 = 15876 \text{ cubic feet}
\]
### Figure 2:
1. **Rectangular Prism Volume** (same as Figure 1):
\[
Volume_{prism} = 13608 \text{ cubic feet}
\]
2. **Triangular Prism Volume**:
- Base = 36 feet
- Height of the triangle = 9 feet
- Height of the prism = 21 feet
- Volume = \(\frac{1}{2} \times \) base \(\times\) height of triangle \(\times\) height of prism = \(\frac{1}{2} \times 36 \times 9 \times 21\)
\[
Volume_{triangular\_prism} = \frac{1}{2} \times 36 \times 9 \times 21 = \frac{1}{2} \times 36 \times 189 = 36 \times 94.5 = 3402 \text{ cubic feet}
\]
3. **Total Volume of Figure 2**:
\[
Volume_{figure2} = Volume_{prism} + Volume_{triangular\_prism} = 13608 + 3402 = 17010 \text{ cubic feet}
\]
### Comparing the volumes of the figures:
- \(Volume_{figure1} = 15876\) cubic feet
- \(Volume_{figure2} = 17010\) cubic feet
\[
Volume_{figure2} - Volume_{figure1} = 17010 - 15876 = 1344 \text{ cubic feet}
\]
### Conclusion:
The volume of Figure 2 is greater than the volume of Figure 1, and it is 1344 cubic feet greater. Thus, the correct statement is:
**The volume of figure 2 is 1,344 feet cubed greater than the volume of figure 1.**