Question

Three water storage containers have the same volume in the shape of a cylinder, a cone, and a sphere. The base radius of the cylinder, the base radius of the cone, and the radius of the sphere are equal.

The ratio of the height of the cone to the height of the cylinder is

The ratio of the height of the cone to radius of the sphere is

Answer 1

- The ratio of the height of the cone to the height of the cylinder is 3:1

- The ratio of the height of the cone to radius of the sphere is 4:1

Explanation:

Note that the radii of all the structures are the same.

Let the height of the cone be H and the height of the cylinder be h

The volume of each of the shapes are given below as

Volume of cylinder = πr²h

Volume of a cone = (1/3)πr²H

Volume of a sphere = (4/3)πr³

1) The ratio of the height of the cone to the height of the cylinder

To obtain this, we equate the volume of those two structures

Volume of the cone = Volume of the cylinder

(1/3)πr²H = πr²h

πr² cancels out on both sides and we're left with

(H/3) = h

H = 3h

(H/h) = (3/1)

So, The ratio of the height of the cone to the height of the cylinder is 3:1

2) The ratio of the height of the cone to radius of the sphere

Similarly equating the volumes of the cone and the sphere

(1/3)πr²H = (4/3)πr³

(1/3)πr² cancels out on both sides and we're left with

H = 4r

(H/r) = (4/1)

The ratio of the height of the cone to radius of the sphere is 4:1

Hope this Helps!!!