Question

select the correct answer.
the circle is rotated around the horizontal dashed line. which type of geometrical figure will be generated?
a. cylinder
b. cylindrical shell
c. sphere
d. ring

Explanation:

Step1: Recall the properties of geometric solids formed by rotation

When a circle is rotated around an axis (in this case, the horizontal dashed line), we consider the 3 - D shape formed.

• A cylinder is formed by rotating a rectangle around one of its sides.
• A cylindrical shell is a more complex shape (difference between two cylinders).
• A ring is a 2 - D or 3 - D shape (hollow in the middle) but not formed by a single - circle rotation.
• When a circle is rotated about an axis, every point on the circle moves in a circular path. The set of all these points forms a sphere. Mathematically, if we consider a circle with equation \((x - a)^2+(y - b)^2 = r^2\) and rotate it about an axis (using the method of generating a surface of revolution), using the distance formula and the concept of rotation in 3 - D space. For a circle of radius \(r\) and center \((h,k)\) (in 2 - D), when rotated about an axis (assuming the axis is at a distance such that the circle is fully rotated around it), the equation of the sphere in 3 - D (using the Pythagorean theorem in 3 - D) \(x^{2}+y^{2}+z^{2}=R^{2}\) (where \(R\) is the radius of the sphere formed by the rotation of the circle) can be derived.

Answer:

C. sphere