A cone and a cylinder have the same base area. The height of these two solids is the same. If a sphere has the same radius as that of the cone, then what is the ratio of the volumes of the cone, the sphere and the cylinder?

Since the cone and the cylinder have the same base area, therefore, their radii are the same.

Let the radius of cone and cylinder be r.

Radius of cone = Radius of cylinder = r

It is also given that, the height and radius of these two solids are the same.

∴ Height of the cone = Height of the cylinder = r

Given, the sphere has the same radius as that of the cone. Therefore, radius of the sphere is r.

Volume of the cone $=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}=\frac{1}{3}{\mathrm{\pi r}}^3$ 

Volume of the sphere  $=\frac{4}{3}{\mathrm{\pi r}}^3$

Volume of the cylinder = π(Radius)²× Height = πr³

Ratio of their volumes = $\frac{1}{3}{\mathrm{\pi r}}^3:\frac{4}{3}{\mathrm{\pi r}}^3:{\mathrm{\pi r}}^3$= 1: 4: 3

Hence, the ratio of volumes of the cone, the sphere and the cylinder is 1: 4: 3.