Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.

Given:

Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melted to form a single solid sphere.

To do:

We have to find the radius of the resulting sphere.

Solution:

The radius of the 1st metallic sphere $= 6\ cm$

This implies,

Volume of the 1st metallic sphere $= \frac{4}{3} \pi (6)^3\ cm^3$

The radius of the 2nd metallic sphere $= 8\ cm$

This implies,

Volume of the 2nd metallic sphere $= \frac{4}{3} \pi(8)^3\ cm^3$

The radius of the 3rd metallic sphere $= 10\ cm$

This implies,

Volume of the 3rd metallic sphere $= \frac{4}{3} \pi(10)^3\ cm^3$

Volume of all three metallic spheres $= \frac{4}{3} \pi(6^3+8^3+10^3)\ cm^3$

Let the three spheres be melted and recast into a new metallic sphere of radius $r$

Therefore,

Volume of the new metallic sphere $= \frac{4}{3} \pi r^3$

This implies,

$\frac{4}{3} \pi(6^{3}+8^{3}+10^{3})=\frac{4}{3} \pi r^{3}$

$6^{3}+8^{3}+10^{3}=r^{3}$

$216+512+1000=r^{3}$

$r^3=1728$

$r=\sqrt[3]{1728}$

$r=12 \mathrm{~cm}$

The radius of the resulting sphere is $12 \mathrm{~cm}$.

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Updated on: 10-Oct-2022

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