Question
According to Cavalieri's Principle, which pair of shapes would have equal volumes?
Question 2 options:
A cylinder and a sphere with the same radius.
A cylinder and an oblique cylinder with equal base areas and equal heights.
A cone and a pyramid with equal heights.
Question 2 options:
A cylinder and a sphere with the same radius.
A cylinder and an oblique cylinder with equal base areas and equal heights.
A cone and a pyramid with equal heights.
Answers
GPT-4o mini
Cavalieri's Principle states that if two solids have the same height and the same cross-sectional area at every level, then they have the same volume.
Let's evaluate the given options:
1. **A cylinder and a sphere with the same radius**: These shapes do not have the same volume. The volume of a cylinder is given by \( V = \pi r^2 h \), while the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). Even if they share the same radius, their heights (for the cylinder) or their geometric properties will result in different volumes.
2. **A cylinder and an oblique cylinder with equal base areas and equal heights**: According to Cavalieri's Principle, the volume will indeed be the same as long as the bases are equal and the heights are equal, regardless of the orientation of the cylinder (whether it's straight or oblique).
3. **A cone and a pyramid with equal heights**: This option needs more information. If the pyramid has a base area equal to the base area of the cone (which has a circular base), then they can have equal volumes. However, without that condition being explicitly stated, you cannot assume they have equal volumes.
Based on this analysis, the best option according to Cavalieri's Principle is:
**A cylinder and an oblique cylinder with equal base areas and equal heights.**
Let's evaluate the given options:
1. **A cylinder and a sphere with the same radius**: These shapes do not have the same volume. The volume of a cylinder is given by \( V = \pi r^2 h \), while the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). Even if they share the same radius, their heights (for the cylinder) or their geometric properties will result in different volumes.
2. **A cylinder and an oblique cylinder with equal base areas and equal heights**: According to Cavalieri's Principle, the volume will indeed be the same as long as the bases are equal and the heights are equal, regardless of the orientation of the cylinder (whether it's straight or oblique).
3. **A cone and a pyramid with equal heights**: This option needs more information. If the pyramid has a base area equal to the base area of the cone (which has a circular base), then they can have equal volumes. However, without that condition being explicitly stated, you cannot assume they have equal volumes.
Based on this analysis, the best option according to Cavalieri's Principle is:
**A cylinder and an oblique cylinder with equal base areas and equal heights.**