Question
Which best describes the relationship among the volumes of hemisphere X, cylinder Y, and cone Z?
The sum of the volumes of X and Y equals the volume of Z.
The sum of the volumes of X and Y equals the volume of Z.
The difference of the volumes of X and Z equals the volume of Y.
The difference of the volumes of X and Z equals the volume of Y.
The difference of the volumes of Z and Y equals the volume of X.
The difference of the volumes of Z and Y equals the volume of X.
The sum of the volumes of X and Z equals the volume of Y.
The sum of the volumes of X and Y equals the volume of Z.
The sum of the volumes of X and Y equals the volume of Z.
The difference of the volumes of X and Z equals the volume of Y.
The difference of the volumes of X and Z equals the volume of Y.
The difference of the volumes of Z and Y equals the volume of X.
The difference of the volumes of Z and Y equals the volume of X.
The sum of the volumes of X and Z equals the volume of Y.
Answers
Answered by
GPT-4o mini
To analyze the volumes of a hemisphere (X), a cylinder (Y), and a cone (Z), let's recall the formulas for each shape's volume:
1. **Volume of a Hemisphere** (radius \( r \)):
\[
V_X = \frac{2}{3} \pi r^3
\]
2. **Volume of a Cylinder** (radius \( r \) and height \( h \)):
\[
V_Y = \pi r^2 h
\]
3. **Volume of a Cone** (radius \( r \) and height \( h \)):
\[
V_Z = \frac{1}{3} \pi r^2 h
\]
Now we can compare these volumes under different conditions.
1. **Analyzing the difference between the volumes of cone Z and cylinder Y**:
\[
V_Y - V_Z = \pi r^2 h - \frac{1}{3} \pi r^2 h = \frac{2}{3} \pi r^2 h
\]
This is not equal to \( V_X \).
2. **Analyzing the sum of the volumes of hemisphere X and cylinder Y**:
\[
V_X + V_Y = \frac{2}{3} \pi r^3 + \pi r^2 h
\]
This does not simplify in a way that is equal to either of the other volumes under general conditions.
3. **Analyzing the sum of the volumes of hemisphere X and cone Z**:
\[
V_X + V_Z = \frac{2}{3} \pi r^3 + \frac{1}{3} \pi r^2 h
\]
This is again not in a simple relationship to cylinder Y's volume.
4. **Comparing volumes**:
- The volume of the cylinder \( V_Y \) becomes larger than both \( V_X \) and \( V_Z \) when \( h \) is sufficiently large compared to the radius \( r \).
- The volume of the cone \( V_Z \) is exactly one-third of the volume of a cylinder with the same base and height.
Considering the relationships given, the closest relationship that might hold true under specific contexts is that:
- The difference of the volumes of Z and Y does not equal the volume of X unless specific dimensional ratios are used.
Without specific height or radius values to give a definite relationship among the volumes, none of the statements provided necessarily describe their relationship in all cases. However, there is a common connection in geometry that could relate them, such as specific proportions based on dimensions.
For a targeted examination for an answer, you may need additional information about the dimensions relating these three shapes.
Given the calculations and recalling common geometric properties, the provided statements can't be universally true without additional constraints. The best contextually accurate statement must be revisited with specific values.
1. **Volume of a Hemisphere** (radius \( r \)):
\[
V_X = \frac{2}{3} \pi r^3
\]
2. **Volume of a Cylinder** (radius \( r \) and height \( h \)):
\[
V_Y = \pi r^2 h
\]
3. **Volume of a Cone** (radius \( r \) and height \( h \)):
\[
V_Z = \frac{1}{3} \pi r^2 h
\]
Now we can compare these volumes under different conditions.
1. **Analyzing the difference between the volumes of cone Z and cylinder Y**:
\[
V_Y - V_Z = \pi r^2 h - \frac{1}{3} \pi r^2 h = \frac{2}{3} \pi r^2 h
\]
This is not equal to \( V_X \).
2. **Analyzing the sum of the volumes of hemisphere X and cylinder Y**:
\[
V_X + V_Y = \frac{2}{3} \pi r^3 + \pi r^2 h
\]
This does not simplify in a way that is equal to either of the other volumes under general conditions.
3. **Analyzing the sum of the volumes of hemisphere X and cone Z**:
\[
V_X + V_Z = \frac{2}{3} \pi r^3 + \frac{1}{3} \pi r^2 h
\]
This is again not in a simple relationship to cylinder Y's volume.
4. **Comparing volumes**:
- The volume of the cylinder \( V_Y \) becomes larger than both \( V_X \) and \( V_Z \) when \( h \) is sufficiently large compared to the radius \( r \).
- The volume of the cone \( V_Z \) is exactly one-third of the volume of a cylinder with the same base and height.
Considering the relationships given, the closest relationship that might hold true under specific contexts is that:
- The difference of the volumes of Z and Y does not equal the volume of X unless specific dimensional ratios are used.
Without specific height or radius values to give a definite relationship among the volumes, none of the statements provided necessarily describe their relationship in all cases. However, there is a common connection in geometry that could relate them, such as specific proportions based on dimensions.
For a targeted examination for an answer, you may need additional information about the dimensions relating these three shapes.
Given the calculations and recalling common geometric properties, the provided statements can't be universally true without additional constraints. The best contextually accurate statement must be revisited with specific values.