How do you find the volume of the largest right circular cone that can be inscribed in a sphere of radius r?

Question

Isaiah Allen · Accepted Answer

You can see my answer to the same problem at this link: https://tutor.hix.ai The only different thing is that the radius there was #2# and here #r#, but I made the substituition only at the end.

Luke Alvarez · Answer

undefined To find the volume of the largest right circular cone that can be inscribed in a sphere of radius r, you can use optimization techniques.

The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone.

Given that the cone is inscribed in a sphere, its base will be a circle that lies on the sphere's surface. The radius of this circle will be the same as the sphere's radius, which is r.

Let's denote the height of the cone as h and the radius of the base as r. By using similar triangles, we can find a relationship between the radius of the base and the height of the cone.

The cone's height (h) plus the radius of its base (r) equals the sphere's radius (r). Therefore, h + r = r, or h = 0.

Since the height of the cone is 0, the volume of the cone will also be 0.

Therefore, the largest right circular cone that can be inscribed in a sphere of radius r has a volume of 0.