Question

A sphere of the radius $6\text{cm}$ is melted and made three spheres of radius $3\text{cm},4\text{cm},\text{and}r\text{cm}$ respectively, then the value of $r$ is ______ .

Answer 1

Calculate the value of $r$:

Let the volume of a sphere that is melted be $V$ and the volumes of three small spheres be $V_1,V_2,\text{and}{V}_3$ respectively.

The volume of a sphere is calculated by,

$\mathit{V}\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}\mathit{\pi }{\mathit{r}}^{\mathbf{3}}$, where $V$ is the volume and $r$ is the radius of a sphere.

As the sphere has been melted into three smaller spheres then the volume of the bigger sphere will be equal to the sum of the volume of three smaller spheres.

$$\begin{gathered} \therefore V=V_1+V_2+V_3 \\ \Rightarrow \frac{4}{3}\pi {\left(6\right)}^3=\frac{4}{3}\pi {\left(3\right)}^3+\frac{4}{3}\pi {\left(4\right)}^3+\frac{4}{3}\pi {\left(r\right)}^3 \\ \Rightarrow 6^3=3^3+4^3+r^3 \\ \Rightarrow r^3=216-27-64 \\ \Rightarrow r^3=125 \\ \Rightarrow r=5\text{cm} \end{gathered}$$

Final Answer :

Hence, the value of $r$ is $\mathbf{5}\mathbf{}\mathbf{\text{cm}}$.