Twenty-seven spheres of radius ’r’ were melted and one new sphere was formed. What is the radius of this sphere?

According to the question,

There are 27 spheres, each of radius ‘r’.

Now, for 1 sphere:

Radius = r

Volume of this 1 sphere = 4/3 πr³

Then, for 27 spheres, each of radius ‘r’:

Volume = 27 × 4/3 πr³

⇒ Volume = 36 πr³ …(i)

Even if these 27 spheres were melted to make it into one new sphere, the volume remains unaltered, which means

Volume of this new sphere = Volume of 27 spheres.

By equation (i), we can say that

Volume of this new sphere = 36 πr³ …(ii)

Let radius of this new sphere formed by melting twenty-seven spheres of radius ‘r’ be ‘R’.

Now, Volume of this new sphere = 4/3 πR³ …(iii)

Comparing equations (ii) and (iii), we get

4/3 πR³ = 36 πr³

⇒ 4/3 R³ = 36 r³

⇒ 1/3 R³ = 9 r³

⇒ R³ = 3 × 9 r³

⇒ R³ = 27 r³

⇒ R³ = 3³ r³

⇒ R³ = (3r)³

⇒ R = (3r)^3×1/3

⇒ R = 3r

Thus, radius of this new sphere formed is 3r.