Question

Three solid spheres of gold whose radii are $1$ cm, $6$ cm and $8$ cm, respectively are melted into a single solid sphere. Find the radius of the sphere.

Answer 1

Given that,

Three solid spheres have radii $1cm,6cm$ and $8cm$, respectively.

A single solid sphere is made by melting the three spheres.

To find out,

The radius of the new sphere.

Let the radius of the new sphere be $xcm$

We know that, when one solid is constructed by melting one or more objects, the volume remains same.

Hence, volume of new sphere $=$ sum of volumes of the three sphere.

We know that, volume of a sphere $=\frac{4}{3}\pi r^3$

Hence, $\frac{4}{3}\pi x^3=\frac{4}{3}\pi (1{)}^3+\frac{4}{3}\pi (6{)}^3+\frac{4}{3}\pi (8{)}^3$

$\Rightarrow x^3=(1{)}^3+(6{)}^3+(8{)}^3$

$\Rightarrow x^3=1+216+512$

$\Rightarrow x^3=729$

$\Rightarrow x^3=9\times 9\times 9$

$\Rightarrow x^3=9^3$

$\therefore x=9cm$

Hence, the radius of the new sphere is $9cm$.