Question
Sonequa has two containers, one in the shape of a cylinder and the other in the shape of a cone. The two containers have equal radii and equal heights. She investigated the relationship between the volumes of the cone and the cylinder by transferring water between the two containers. Which of the following claims is most likely to be supported using the results of Sonequa's investigation?
Responses
The volume of a cylinder is two times the volume of a cone when the two containers have equal radii and equal heights, because one full cylinder of water completely and exactly filled the cone two times.
Answer A: The volume of a cylinder is two times the volume of a cone when the two containers have equal radii and equal heights, because one full cylinder of water completely and exactly filled the cone two times.
A
The volume of a cylinder is three times the volume of a cone when the two containers have equal radii and equal heights, because one full cylinder of water completely and exactly filled the cone three times.
Answer B: The volume of a cylinder is three times the volume of a cone when the two containers have equal radii and equal heights, because one full cylinder of water completely and exactly filled the cone three times.
B
The volume of a cylinder is two times the volume of a cone when the two containers have equal radii and equal heights, because two full cylinders of water completely and exactly filled the cone one time.
Answer C: The volume of a cylinder is two times the volume of a cone when the two containers have equal radii and equal heights, because two full cylinders of water completely and exactly filled the cone one time.
C
The volume of a cylinder is three times the volume of a cone when the two containers have equal radii and equal heights, because three full cylinders of water completely and exactly filled the cone one time.
Responses
The volume of a cylinder is two times the volume of a cone when the two containers have equal radii and equal heights, because one full cylinder of water completely and exactly filled the cone two times.
Answer A: The volume of a cylinder is two times the volume of a cone when the two containers have equal radii and equal heights, because one full cylinder of water completely and exactly filled the cone two times.
A
The volume of a cylinder is three times the volume of a cone when the two containers have equal radii and equal heights, because one full cylinder of water completely and exactly filled the cone three times.
Answer B: The volume of a cylinder is three times the volume of a cone when the two containers have equal radii and equal heights, because one full cylinder of water completely and exactly filled the cone three times.
B
The volume of a cylinder is two times the volume of a cone when the two containers have equal radii and equal heights, because two full cylinders of water completely and exactly filled the cone one time.
Answer C: The volume of a cylinder is two times the volume of a cone when the two containers have equal radii and equal heights, because two full cylinders of water completely and exactly filled the cone one time.
C
The volume of a cylinder is three times the volume of a cone when the two containers have equal radii and equal heights, because three full cylinders of water completely and exactly filled the cone one time.
Answers
GPT-4o mini
To determine the correct relationship between the volumes of the cylinder and the cone when they have equal radii and heights, we start by recalling the volume formulas for both shapes:
1. The volume \( V_{cylinder} \) of a cylinder is given by:
\[
V_{cylinder} = \pi r^2 h
\]
2. The volume \( V_{cone} \) of a cone is given by:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Given that the radius \( r \) and height \( h \) are the same for both containers, we can express the relationship between their volumes:
\[
V_{cylinder} = \pi r^2 h
\]
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
To find the ratio of the volume of the cylinder to the volume of the cone, we can divide the two volumes:
\[
\text{Ratio} = \frac{V_{cylinder}}{V_{cone}} = \frac{\pi r^2 h}{\frac{1}{3} \pi r^2 h} = 3
\]
This shows that the volume of the cylinder is three times the volume of the cone.
Now let's analyze the claims:
- **Answer A:** This claim states that the volume of a cylinder is two times the volume of a cone. This is incorrect as we've established the ratio is 3:1.
- **Answer B:** This claim states that the volume of a cylinder is three times the volume of a cone because one full cylinder completely filled the cone three times. This is correct, as we know that the cylinder's volume is three times that of the cone.
- **Answer C:** This claim also suggests that the volume of a cylinder is two times that of a cone. This is incorrect for the same reason as Answer A.
- **Answer D:** This claim states that the cylinder's volume is three times that of a cone, but the reasoning provided does not align correctly with the fact established, as it should refer to one full cylinder filling the cone one time, not three.
Based on this analysis, the claim most likely to be supported from Sonequa's investigation is:
**Answer B:** The volume of a cylinder is three times the volume of a cone when the two containers have equal radii and equal heights, because one full cylinder of water completely and exactly filled the cone three times.
1. The volume \( V_{cylinder} \) of a cylinder is given by:
\[
V_{cylinder} = \pi r^2 h
\]
2. The volume \( V_{cone} \) of a cone is given by:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Given that the radius \( r \) and height \( h \) are the same for both containers, we can express the relationship between their volumes:
\[
V_{cylinder} = \pi r^2 h
\]
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
To find the ratio of the volume of the cylinder to the volume of the cone, we can divide the two volumes:
\[
\text{Ratio} = \frac{V_{cylinder}}{V_{cone}} = \frac{\pi r^2 h}{\frac{1}{3} \pi r^2 h} = 3
\]
This shows that the volume of the cylinder is three times the volume of the cone.
Now let's analyze the claims:
- **Answer A:** This claim states that the volume of a cylinder is two times the volume of a cone. This is incorrect as we've established the ratio is 3:1.
- **Answer B:** This claim states that the volume of a cylinder is three times the volume of a cone because one full cylinder completely filled the cone three times. This is correct, as we know that the cylinder's volume is three times that of the cone.
- **Answer C:** This claim also suggests that the volume of a cylinder is two times that of a cone. This is incorrect for the same reason as Answer A.
- **Answer D:** This claim states that the cylinder's volume is three times that of a cone, but the reasoning provided does not align correctly with the fact established, as it should refer to one full cylinder filling the cone one time, not three.
Based on this analysis, the claim most likely to be supported from Sonequa's investigation is:
**Answer B:** The volume of a cylinder is three times the volume of a cone when the two containers have equal radii and equal heights, because one full cylinder of water completely and exactly filled the cone three times.