A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of two of these balls are 1.5 cm and 2 cm. The radius of the third ball is :

1. 0.5 cm

2. 1 cm

3. 1.5 cm

4. 2.5 cm

Radius of larger spherical metallic ball, R = 3 cm

Radius of smaller spherical balls are 1.5 cm, 2 cm and r cm

Given,

A spherical metallic ball of radius 3 cm is melted and recast into three spherical balls.

∴ Volume of larger spherical ball = Volume of ball of radius 1.5 cm + Volume of ball of radius 2 cm + Volume of ball of radius r cm

$\Rightarrow \dfrac{4}{3}π\text{R}^3 = \dfrac{4}{3}π \times 1.5^3 + \dfrac{4}{3}π \times 2^3 + \dfrac{4}{3}π\text{r}^3 \\[1em] \Rightarrow \dfrac{4}{3}π\text{R}^3 = \dfrac{4}{3}π(1.5^3 + 2^3 + \text{r}^3) \\[1em] \Rightarrow \text{R}^3 = (1.5^3 + 2^3 + \text{r}^3) \\[1em] \Rightarrow 3^3 = 3.375 + 8 + \text{r}^3 \\[1em] \Rightarrow \text{r}^3 = 27 - 3.375 - 8 \\[1em] \Rightarrow \text{r}^3 = 15.625 \\[1em] \Rightarrow \text{r} = \sqrt[3]{15.625} \\[1em] \Rightarrow \text{r} = 2.5 \text{ cm.}$

Hence, option 4 is the correct option.