A sphere is split in the ratio 1 : 3. The larger part is molded into a cone having a height equal to the radius of its base, while the smaller part is molded into a cylinder having a height equal to the radius of its base. What would be the ratio of the radius of the base of the cone to the height of the cylinder?

Let 4V be the volume of the sphere. It is split in ratio of 1:3 , so smaller one has volume = V and larger part has volume = 3V.

The larger part is molded into a cone.

3V = $$\frac{1}{3} \pi r^2 h$$ = $$\frac{1}{3} \pi r^3$$

The smaller part is modeled into a cylinder

V = $$\pi r^2h$$ => V =  $$\pi r^3$$

Ratio of the radius of the base of the cone to the height of the cylinder

= $$\sqrt[3]{ \frac{3V \times 3}{V} }$$ = $$ \sqrt[3]{9} : 1 $$

So, the answer would be option d)$$ \sqrt[3]{9} : 1 $$.