A sphere of diameter 12.6 cm is melted and cast into a right circular cone of height 25.2 cm. The radius of the base of the cone is :

1. 2 cm

2. 2.1 cm

3. 3 cm

4. 6.3 cm

Radius of sphere, r = $\dfrac{\text{diameter}}{2} = \dfrac{12.6}{2} = 6.3 \text{ cm.}$

Volume of sphere = $\dfrac{4}{3}π\text{r}^3$

Radius of the cone = R cm

Height of the cone, h = 25.2 cm

Volume of cone = $\dfrac{1}{3}π\text{R}^2 \text{h}$

Since, sphere is melted and recasted into a cone, the volume remains the same.

$\therefore \dfrac{1}{3}π\text{R}^2 \text{h} = \dfrac{4}{3}π\text{r}^3 \\[1em] \Rightarrow \dfrac{1}{3}\text{R}^2 \text{h} = \dfrac{4}{3}\text{r}^3 \\[1em] \Rightarrow \text{R}^2 = \dfrac{4 \times 3}{3 \times \text{h}} \times \text{r}^3 \\[1em] \Rightarrow \text{R}^2 = \dfrac{12}{3 \times 25.2} \times 6.3^3 \\[1em] \Rightarrow \text{R}^2 = \dfrac{12}{75.6} \times 250.047 \\[1em] \Rightarrow \text{R}^2 = \dfrac{3000.564}{75.6} \\[1em] \Rightarrow \text{R}^2 = 39.69 \\[1em] \Rightarrow \text{R} = \sqrt{39.69} \\[1em] \Rightarrow \text{R} = 6.3 \text{ cm.}$

Hence, option 4 is the correct option.