Question

A cylinder whose height is 3 times its radius is inscribed in a cone whose height is 6 times its radius. What fraction of the cone's volume lies inside the cylinder? Express your answer as a common fraction.

Answer 1

To find the fraction of the cone's volume that lies inside the cylinder, we can calculate the volumes of both the cone and the cylinder. The volume of the cylinder is 3πr³ and the volume of the cone is 2πr³. The fraction of the cone's volume inside the cylinder is 3/2.

Step-by-step explanation:

To find the fraction of the cone's volume that lies inside the cylinder, we first need to determine the volumes of both the cone and the cylinder.

The volume of a cylinder is given by the formula V = πr²h.

Since the height of the cylinder is 3 times its radius, we can write it as h = 3r. Thus, the volume of the cylinder is Vc = πr²(3r) = 3πr³.

The volume of a cone is given by the formula V = 1/3πr²h.

Since the height of the cone is 6 times its radius, we can write it as h = 6r. Thus, the volume of the cone is Vn = 1/3πr²(6r) = 2πr³.

Therefore, the fraction of the cone's volume that lies inside the cylinder is given by the ratio of their volumes: Vc/Vn = (3πr³)/(2πr³) = 3/2.

So, the fraction of the cone's volume that lies inside the cylinder is 3/2.