Question
a circle has its center at the origin of a coordinate plane. if you were to rotate that quarter about the x-ais what three dimensional object would be formed
Answers
GPT-4o mini
When you rotate a quarter circle about the x-axis, you create a three-dimensional shape known as a **circular cone**. The quarter circle is defined in the first quadrant of the coordinate plane, bounded by the x-axis and y-axis.
1. **Quarter circle definition**: The quarter circle has a radius \( r \) and is part of the circle defined by the equation \( x^2 + y^2 = r^2 \), for \( x \geq 0 \) and \( y \geq 0 \).
2. **Rotation**: When this quarter circle is rotated about the x-axis, every point on the curve sweeps out a surface in three-dimensional space. The resulting shape has a circular cross-section perpendicular to the x-axis at any point.
3. **Resulting shape**: The resulting solid is a cone with its apex at the origin (0,0,0) and its base on the plane perpendicular to the x-axis at \( x = r \).
So, the object formed by rotating a quarter circle about the x-axis is a circular cone.
1. **Quarter circle definition**: The quarter circle has a radius \( r \) and is part of the circle defined by the equation \( x^2 + y^2 = r^2 \), for \( x \geq 0 \) and \( y \geq 0 \).
2. **Rotation**: When this quarter circle is rotated about the x-axis, every point on the curve sweeps out a surface in three-dimensional space. The resulting shape has a circular cross-section perpendicular to the x-axis at any point.
3. **Resulting shape**: The resulting solid is a cone with its apex at the origin (0,0,0) and its base on the plane perpendicular to the x-axis at \( x = r \).
So, the object formed by rotating a quarter circle about the x-axis is a circular cone.
options are
a cylinder
a hemisphere
a cone
a sphere
a cylinder
a hemisphere
a cone
a sphere
GPT-4o mini
When you rotate a quarter circle about the x-axis, the correct option from your list is **a cone**.
Here's a summary of the reasoning:
- A **cylinder** would result from rotating a rectangle, not a quarter circle.
- A **hemisphere** is half of a sphere, which would require rotating a half-circle, not a quarter circle.
- A **sphere** would result from rotating a full circle, not a quarter circle.
- A **cone** is the shape formed by rotating a quarter circle (in the first quadrant) around the x-axis.
Thus, the answer is **a cone**.
Here's a summary of the reasoning:
- A **cylinder** would result from rotating a rectangle, not a quarter circle.
- A **hemisphere** is half of a sphere, which would require rotating a half-circle, not a quarter circle.
- A **sphere** would result from rotating a full circle, not a quarter circle.
- A **cone** is the shape formed by rotating a quarter circle (in the first quadrant) around the x-axis.
Thus, the answer is **a cone**.
are you sure
GPT-4o mini
Yes, I am sure. The correct answer is indeed **a cone** when you rotate a quarter circle about the x-axis.
Here's a brief recap to clarify:
1. **Quarter Circle**: Consider a quarter of a circle in the first quadrant, with its center at the origin (0,0) and a radius \( r \). This quarter circle extends from (0,0) to (r,0) along the x-axis and from (0,0) to (0,r) along the y-axis.
2. **Rotation About the X-Axis**: When this quarter circle is rotated around the x-axis, each point on the arc forms a circular path in three-dimensional space.
3. **Shape Formed**: The shape that results from this rotation is a cone with:
- A circular base whose radius is equal to the radius of the quarter circle (i.e., \( r \)).
- The apex of the cone at the origin (0,0,0) and extending outwards along the x-axis.
Thus, from the provided options:
- **Cylinder**: Not correct.
- **Hemisphere**: Not correct.
- **Cone**: Correct.
- **Sphere**: Not correct.
So the answer is **a cone**.
Here's a brief recap to clarify:
1. **Quarter Circle**: Consider a quarter of a circle in the first quadrant, with its center at the origin (0,0) and a radius \( r \). This quarter circle extends from (0,0) to (r,0) along the x-axis and from (0,0) to (0,r) along the y-axis.
2. **Rotation About the X-Axis**: When this quarter circle is rotated around the x-axis, each point on the arc forms a circular path in three-dimensional space.
3. **Shape Formed**: The shape that results from this rotation is a cone with:
- A circular base whose radius is equal to the radius of the quarter circle (i.e., \( r \)).
- The apex of the cone at the origin (0,0,0) and extending outwards along the x-axis.
Thus, from the provided options:
- **Cylinder**: Not correct.
- **Hemisphere**: Not correct.
- **Cone**: Correct.
- **Sphere**: Not correct.
So the answer is **a cone**.