The area of the base of a cone is $$64\pi  cm^2$$ while its slant height is 17 cm. This cone is remolded to obtain a solid sphere. The radius of this sphere will be:

According to the question, the cone is remolded to a solid sphere.

So, the volume of sphere will be equal to the volume of cone.

Volume of cone:

Radius of the cone, r = 8 cm (as $$\pi r^2 = 64 \pi)$$

Height of cone, $$h = \sqrt{17^2 - 8^2} = 15 cm$$

so, volume of cone =  $$\frac{1}{3}(\pi r^2 h)$$

$$=\frac{1}{3} (\pi \times 8 \times 8 \times15)$$

= $$320 \pi$$

Volume of sphere:

Let radius of sphere=R

So, volume of sphere = $$\frac{4}{3}(\pi R^3)$$

We know, volume of sphere=volume of cone

or, $$\frac{4}{3}(\pi R^3) = 320 \pi$$

or, $$R^3 = 240$$   or, $$R = \sqrt[3]{8} \times 30$$   or, $$R = 2\sqrt[3]{30}$$