Question
Two cylinders each with a height of 50 meters are shown. Which statements about cylinders P and S are TRUE? Select ALL true statements.
right cylinder P with radius x. oblique cylinder S with radius y
Question 11 options:
If x = y, the volume of cylinder P is greater than the volume of cylinder S, because cylinder P is a right cylinder.
If x = y, the volume of cylinder P is equal to the volume of cylinder S, because the cylinders are the same height.
If x = y, the volume of cylinder P is less than the volume of cylinder S, because cylinder S is slanted.
If x < y, the area of a horizontal cross section of cylinder P is less than the area of a horizontal cross section of cylinder S.
right cylinder P with radius x. oblique cylinder S with radius y
Question 11 options:
If x = y, the volume of cylinder P is greater than the volume of cylinder S, because cylinder P is a right cylinder.
If x = y, the volume of cylinder P is equal to the volume of cylinder S, because the cylinders are the same height.
If x = y, the volume of cylinder P is less than the volume of cylinder S, because cylinder S is slanted.
If x < y, the area of a horizontal cross section of cylinder P is less than the area of a horizontal cross section of cylinder S.
Answers
GPT-4o mini
To determine which statements about cylinders P and S are true, we need to consider the properties of right cylinders and oblique cylinders, particularly regarding their volumes and cross-section areas.
1. **Volume of Right Cylinder (P)**: The volume \( V \) of a right cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
2. **Volume of Oblique Cylinder (S)**: An oblique cylinder also has a volume given by the same formula:
\[
V = \pi r^2 h
\]
The volume is determined by the base area and the height, and not affected by the slant of the cylinder.
Now let's evaluate the statements based on this information:
1. **Statement**: *If x = y, the volume of cylinder P is greater than the volume of cylinder S, because cylinder P is a right cylinder.*
- **False**: If \( x = y \), the volumes of both cylinders are equal, regardless of the orientation.
2. **Statement**: *If x = y, the volume of cylinder P is equal to the volume of cylinder S, because the cylinders are the same height.*
- **True**: Since both cylinders have the same height and radius, their volumes are equal.
3. **Statement**: *If x = y, the volume of cylinder P is less than the volume of cylinder S, because cylinder S is slanted.*
- **False**: As noted above, if \( x = y \), the volumes are equal.
4. **Statement**: *If x < y, the area of a horizontal cross section of cylinder P is less than the area of a horizontal cross section of cylinder S.*
- **True**: The area of a horizontal cross-section of each cylinder is determined by the formula \( A = \pi r^2 \). If \( x < y \), then the area of the cross-section of cylinder P is less than that of cylinder S.
In conclusion, the true statements about cylinders P and S are:
- If x = y, the volume of cylinder P is equal to the volume of cylinder S, because the cylinders are the same height.
- If x < y, the area of a horizontal cross section of cylinder P is less than the area of a horizontal cross section of cylinder S.
1. **Volume of Right Cylinder (P)**: The volume \( V \) of a right cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
2. **Volume of Oblique Cylinder (S)**: An oblique cylinder also has a volume given by the same formula:
\[
V = \pi r^2 h
\]
The volume is determined by the base area and the height, and not affected by the slant of the cylinder.
Now let's evaluate the statements based on this information:
1. **Statement**: *If x = y, the volume of cylinder P is greater than the volume of cylinder S, because cylinder P is a right cylinder.*
- **False**: If \( x = y \), the volumes of both cylinders are equal, regardless of the orientation.
2. **Statement**: *If x = y, the volume of cylinder P is equal to the volume of cylinder S, because the cylinders are the same height.*
- **True**: Since both cylinders have the same height and radius, their volumes are equal.
3. **Statement**: *If x = y, the volume of cylinder P is less than the volume of cylinder S, because cylinder S is slanted.*
- **False**: As noted above, if \( x = y \), the volumes are equal.
4. **Statement**: *If x < y, the area of a horizontal cross section of cylinder P is less than the area of a horizontal cross section of cylinder S.*
- **True**: The area of a horizontal cross-section of each cylinder is determined by the formula \( A = \pi r^2 \). If \( x < y \), then the area of the cross-section of cylinder P is less than that of cylinder S.
In conclusion, the true statements about cylinders P and S are:
- If x = y, the volume of cylinder P is equal to the volume of cylinder S, because the cylinders are the same height.
- If x < y, the area of a horizontal cross section of cylinder P is less than the area of a horizontal cross section of cylinder S.