I think he meant geometrically because he has not yet a working knowledge of calculus since this is in the precalculus section.Quinzio said:You prove it with integration of the sphere sections.
I'm not sure what d'you mean "geometrically".
The formula for finding the volume of a sphere is V = (4/3)πr^3, where π is a constant and r is the radius of the sphere.
The equation of a plane intersecting a sphere can be found by setting the equation of the plane equal to the equation of the sphere and solving for the variable. This will give you the point(s) of intersection.
The volume of a sphere and the plane intersecting it are not directly related. The volume of the sphere will remain the same regardless of how the plane intersects it. The intersection of the sphere and the plane will determine the shape and size of the resulting figure, but not the volume.
Changing the position of the plane can affect the volume of the intersecting region in different ways. If the plane is moved closer to the center of the sphere, it may result in a smaller intersecting region and therefore a smaller volume. However, if the plane is moved further away from the center, it may result in a larger intersecting region and a larger volume.
Yes, a plane can intersect a sphere more than once. Depending on the position and orientation of the plane, it can intersect the sphere at different points, resulting in multiple intersections. The number of intersections will depend on the specific values of the equations for the plane and the sphere.