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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

5 answers

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.

options are
a cylinder
a hemisphere
a cone
a sphere

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.

are you sure

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.